Given a set of integers $g_0 < \ldots < g_e$, how many relations are there among them? This deceptively simple numerical question can be translated in an algebraic one, as it asks for the number of minimal generators of the defining ideal of a semigroup ring. The study of numerical properties of monoids via their associated semigroup ring (and viceversa) is a classic research topic; in this talk we explore a novel approach which extends this problem beyond the realm of semigroup rings to the one of monomial ideals, allowing to use a larger set of techniques and giving general bounds and insight.
Given a point $x$ in a real algebraic variety $X$, which points in space are closer to $x$ than to any other point in $X$? The answer to this question is precisely the so called Voronoi cell of $x$, and the set of all Voronoi cells of points in the variety is called the Voronoi diagram of $X$. Describing the Voronoi diagram of a variety is a fundamental problem in Metric Algebraic Geometry, and it has been previously studied for the Euclidean distance. In this talk we explore the description of the Voronoi diagram of algebraic varieties of codimension one when the distance arises from a polyhedral norm. We will exemplify these diagrams with varieties arising from Algebraic Statistics and optimal transport.
The secondary fan of a polytope stratifies the regular subdivisions of the polytope. We compute this fan for the hypersimplices $\Delta(2,7)$ and $\Delta(3,6)$. We also find new families of rays of the secondary fan of $\Delta(k,n)$ which do not lie on the respective Dressians. In the special case $k=2$ our results are closely related to metric spaces on $n$ points.
This is joint work with Michael Joswig and Lars Kastner.
Pencils of matrices are 2-dimensional linear subspaces in a space of matrices. One can identify the space of squared matrices with an affine open subset of a Grassmannian, and embed a given pencil L into it. In this talk we describe the closure Y_L of the pencil inside the Grassmannian as a blow-up of the complex projective plane at finitely many points, and we relate points in the exceptional locus to a C*-action on the Grassmannian. This is a joint work in progress with F. Gesmundo and H. Keneshlou.
In this talk, we present a characterisation of realizability of matroid quotients, over an infinite field, in terms of realizability of a single matroid associated to it, called the Higgs major. We then present some applications to the relative realizability problem for Bergman Fans in Tropical Geometry.
Toric varieties build an important class in algebraic geometry as their high symmetry allows to translate algebraic geometric properties into combinatorics and vice versa. Constructing toric degenerations of a variety enables us to also give combinatorial descriptions of geometric invariants of (non-toric) varieties.
In this talk we discuss two approaches to obtain toric degenerations: In the first part of the talk we use the Khovanskii bases based approach to obtain toric degenerations of varieties with torus action of complexity one. Afterwards we discuss how to obtain toric degenerations based on the theory of Cox rings.
In this talk, we introduce the concept of Partitioned Independent Component Analysis (PICA), an extension of the classical Independent Component Analysis technique. ICA traditionally aims at separating a mixture of signals into its independent components by determining a mixing matrix. Our work focuses on the conditions under which this mixing matrix can be identified when the assumption of mutual independence among signals is relaxed. Building on recent work of Mesters and Zwiernik, we explore the cases where only subsets of source signals are required to be mutually independent, in other words PICA. Utilizing algebraic techniques similar to previous work, we investigate the identifiability of the mixing matrix in such cases. In this talk, we discuss our findings that reveal that the conditions for identifiability can be generalized, hence broadening the use of ICA in practical cases where traditional independence assumptions may not hold.
This presentation is a mix of two papers: O'Neill's Theorem for PL-Approximations (joint with S. Govindan) and O'Neill's Theorem for Games (joint with S. Govindan and R. Laraki). O'Neill's Theorem in fixed point theory (B. O'Neill, 1953) presents the structure of fixed points of a map around a connected component of fixed points under perturbations of the map. We prove a game-theoretic version of this theorem which shows the structure of Nash-equilibria around a component of Nash-equilibria under payoff perturbations.
Davison proved that the moduli space of objects in a k-linear 2-Calabi--Yau category is formally locally a quiver variety. Bellamy--Schedler gave a classification of which quiver varieties admit symplectic resolutions of singularities, and more recently with Craw classified symplectic resolutions in most cases. It is natural to wonder to what extent these two results could be combined to classify symplectic resolutions of singularities for the moduli space of objects in a 2-Calabi--Yau category. Note that 2-Calabi--Yau categories include the bounded derived category of a K3 surface, the wrapped Fukaya category of a symplectic Liouville 4-manifold, and the category of Higgs bundles on a closed Riemann surface.
In joint work with Travis Schedler, we develop an obstruction theory to extend local resolutions of stratified spaces to global resolutions. The strategy is to (1) choose resolutions around basepoints of minimal strata, (2) extend from a basepoint to the entire stratum, and (3) check compatibility of extensions across strata. The key lemma is a parallel transport type result to extend resolutions along simple exit paths. Then, for each s in S a stratum, parallel transport gives an action of the fundamental group of S on the set of germs of symplectic resolutions at s, which can be interpreted as an obstruction to (2). We prove that monodromy-free, compatible local resolutions extend and glue to a unique global resolution. In other language, the assignment of an open set U to the set of isomorphism classes of symplectic resolutions over of U is an S-constructible sheaf, where S is the stratification in symplectic leaves.
This talk will serve as a gentle introduction to these ideas, highlighting applications and working with small examples like the orbit space of a cyclic group action on a 2-dimensional complex torus.
Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose signature satisfies certain polynomial equations. Particular emphasis lies on the role of the non-associative halfshuffle, which makes it possible to describe varieties of paths that satisfy certain relations all along their trajectory. Specifically, we may understand the set of paths on a given classical algebraic variety in R d starting from a fixed point as a path variety. While halfshuffle varieties are stable under stopping paths at an earlier time, we furthermore study varieties that are stable under concantenation of paths. We point out how the notion of dimension for path varieties crucially depends on the fact that they may be reducible into countably infinitely many subvarieties. Finally, we see that studying halfshuffle varieties naturally leads to a generalization of classical algebraic curves, surfaces and affine varieties in finite dimensional space, where these generalized algebraic sets are now described through iterated-integral equations.
As a highlight for this talk, I will illustrate how we can use this machinery to translate every system of ODEs with polynomial coefficients into a purely algebraic description of the variety of solutions to the ODE system.
To every simple toric ideal I_T one can associate the strongly robust simplicial complex Δ_T, which determines the strongly robust property for all ideals that have I_T as their bouquet ideal. We show that for the simple toric ideals of monomial curves in A^s, the strongly robust simplicial complex Δ_T is either {∅} or contains exactly one 0-dimensional face. In the case of monomial curves in A^3, the strongly robust simplicial complex Δ_T contains one 0-dimensional face if and only if the toric ideal I_T is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way.
In this talk, we'll introduce a new method for computing the kernel of a polynomial map which is homogeneous with respect to a multigrading. We first demonstrate how to quickly compute a matrix of maximal rank for which the map has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gröbner basis techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is embarassingly parallel. This is joint work with Joseph Cummings.
Every symmetric convex body induces a norm on its affine hull. The object of our study is the bisector of two points with respect to this norm. A topological description of bisectors is known in the 2 and 3-dimensional cases and recent work of Criado, Joswig and Santos (2022) expanded this to a fuller characterisation of the geometric, combinatorial and topological properties of the bisector. A key object introduced was the bisection fan of a polytope which they were able to explicitly describe in the case of the tropical norm. We discuss the bisector as a polyhedral complex, introduce the notion of bisection cones and give combinatorial descriptions of the bisection fan corresponding to other polyhedral norms. This is joint work with Katharina Jochemko.
A positive geometry consists of a real projective variety and a semialgebraic subset (its “positive part”), together with a canonical rational form which satisfies a recursive definition when restricted to the boundary of the semialgebraic set. Positive geometries have been objects of interest in physics, and have recently started being explored mathematically. In my talk, I will focus on hyperplane arrangements in projective space. Regions in a hyperplane arrangement complement are polytopes, which are known to be positive geometries. I will discuss when such a region remains a positive geometry after taking the wonderful compactification of the arrangement. This talk is based on work in progress with S. Brauner, C. Eur, and L. Pratt.
Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. We construct a family of non-hyperelliptic surfaces of genus at least 3 where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an M-curve). I will also share some numerical approximations where we try to detect the underlying algebraic curve through sampling. This is based on joint work with Ángel David Ríos Ortiz.
Weighted finite automata (WFAs), with coefficients in a semiring K, are a basic computational model that generalizes linear recurrence sequences to a (noncommutative) multivariate setting. A classical theorem of Schützenberger from the 60s characterizes their generating series as precisely the noncommutative rational series; from another perspective, many problems about weighted automata are related to understanding the dynamics of a vector under the action of a finitely generated matrix semigroup. After discussing the basic notions, I will restrict to the case where K is a field, illustrate a recent new invariant (the linear hull), and state a theorem that characterizes rational series generated by unambiguous, respectively, deterministic WFAs by an arithmetical property on their coefficients (the univariate case is already due to Pólya; the multivariate case resolves a conjecture of Reutenauer from the 70s). Together with a computability result on the (linear) Zariski closure of a finitely generated matrix semigroup, this shows the algorithmic decidability of the determinizability and the unambigualizability problems for WFAs with coefficients in a field. I will also discuss some related open problems.
(Joint work with J. Bell)
Kempe equivalence is a classical and important notion on vertex coloring in graph theory.
In this talk, I introduce several ideals associated with graphs and provide a method determine whether two k-colorings are Kempe equivalent via commutative algebra. Moreover, I give a way to compute all k-colorings of a graph up to Kempe equivalence by virtue of the algebraic technique on Gröbner bases.
This talk is based on joint work with Hidefumi Ohsugi.
Given two rational univariate polynomials, the Wasserstein distance of their associated measures is an algebraic number. We determine the algebraic degree of the squared Wasserstein distance, serving as a measure of algebraic complexity of the corresponding optimization problem. The computation relies on the structure of a subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation.
Positroids are a class of nice realisable matroids, in bijection with many different combinatorial objects. In this talk I will introduce a new characterisation of positroids in terms of essential sets, which allow for a more intuitive understanding of their structure. I will show how one can use essential sets to study positroids and their cells inside the non-negative Grassmannian.
In general, the set of associated primes of an ideal changes when looking at powers of the ideal. These changes have been studied in many different settings. In the Noetherian case, it is well known that the sequence of associated primes of powers of an ideal stabilizes.
It is not known when this stabilization occurs; however, some classes of ideals are well understood. For example, the associated primes of powers of edge ideals of finite simple graphs can be fully described by the structure of the graph. This talk focuses on the stabilization of associated primes of powers of monomial ideals and presents a technique to develop upper bounds for the power of an ideal after which the sequence is non-increasing. This approach is based on describing membership of monomials in ideals via solutions of systems of linear inequalities.
Based on a joint work with Roswitha Rissner and Clemens Heuberger.
We investigate the problem of representing a Borel measure supported on an elliptic normal curve, when restricted to bounded degree polynomials, as the sum of Dirac measures. The smallest number of Dirac measures needed to represent any such measure is called the Carathéodory number. This number governs the complexity of cubature rules and can be interpreted as the rank of Waring-type minimal representations with nonnegative coefficients.
Despite its importance, and several asymptotic results, no exact values for the Carathéodory numbers were known beyond the rational case. In this talk, we show how in the genus one case this number depends on the topology of the real locus of the supporting curve, exploiting the duality with nonnegative polynomials.
Based on a joint work with Greg Blekherman and Rainer Sinn.
Geometric topologists like to study spaces of arbitrary dimensions. Fortunately, we at least limit ourselves to studying manifolds, which locally mimic Euclidean space. Dimension four forms a "phase transition" between low- and high-dimensional manifolds, exhibiting unique behaviour and necessitating bespoke tools. I will describe the source of this curious phenomenon, giving a few guiding examples and constructions. The key source of the problem or appeal, depending on your perspective, of 4-dimensional manifolds turns out to be the difficulty in embedding surfaces therein.
In this talk we introduce modular forms and harmonic weak Maass forms, real-analytic generalizations of holomorphic modular forms. We present some applications of the theory in number theory and to the theory of elliptic curves.
The classical game theory notion of Nash equilibrium imposes on the players the tacit assumption of acting independently from each other. However, in real-life situations this might not at all be a natural restriction. In 2003, the philosopher Wolfgang Spohn introduced the concept of dependency equilibrium (DE) which allows cooperation of the players. His definition leads to a system of equations in many real variables involving rational expressions and limits.
We try to handle these equations employing tools from algebraic geometry. Among other things, we show that the games whose set of DE equals the non-negative real part of the Spohn variety, an algebraic variety recently introduced by Portakal and Sturmfels, form a Zariski open set in the affine space of all games of a fixed size. We explicitly determine this set for games with two players who have two pure strategies each, and we prove that, in general, the Spohn variety contains its real points as a Zariski dense set.
This is joint work with Irem Portakal.
Mirror symmetry gives a correspondence between certain Fano varieties and Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. I will present a set of combinatorial conditions Phi on pairs of Laurent polynomials (f,g) which imply the existence of mirror Fano varieties X_f and X_g related by a blow-up map X_g \to X_f. These criteria generalise the relationship between fans of toric varieties related by toric blowup; I will explain how in some key examples. Time permitting, I will discuss a new approach to constructing mirrors to Laurent polynomials, which is the main idea in the proof that Laurent polynomials in two variables satisfying the conditions Phi have mirrors related by blowing up in one point. This is based on upcoming joint work with Mark Gross.
The problem of deciding and certifying membership in polynomial ideals is of fundamental importance in Computer Algebra and its applications. The complexity of this and the related problem of Gröbner basis computation has been studied (at least) since the 80s, with scary "doubly-exponential" worst-case examples by Mayr & Meyer.
The similar problem of membership in subalgebras of the polynomial ring is in some way (and with many caveats) parallel to that of ideal membership, with SAGBI bases playing the role of Gröbner bases. We investigate the computational complexity of this problem for general, homogeneous, monomial and univariate subalgebras and compare it to the ideal situation.
This is work in progress.
(This talk does NOT assume any prior knowledge of matroids!)
Symmetric edge polytopes are a class of reflexive lattice polytopes depending on the combinatorial data of a graph. Such objects arise in many different contexts, including finite metric space theory, physics and optimal transport, and have been studied extensively in the last few years.
The aim of this talk is to show that symmetric edge polytopes are special instances of a more general construction that associates a reflexive lattice polytope with every regular matroid. A matroid is called regular if it can be represented over every field; by work of Tutte, a matroid is regular if and only if it can be represented by a totally unimodular matrix, i.e. a matrix whose square submatrices of any size all have determinant equal to -1, 0 or 1.
We will show that regular matroids are the right framework for studying symmetric edge polytopes, as two (classical) symmetric edge polytopes turn out to be unimodularly equivalent precisely when the two associated graphs give rise to the same graphic matroid up to isomorphism.
This is joint work with Martina Juhnke-Kubitzke and Melissa Koch.
Relationships between matroids and the permutahedral toric variety are central to matroid Hodge theory. One might wish to generalise these relationships, and the Hodge theory, to delta-matroids, which are Coxeter type B objects. I'll introduce delta-matroids and present one such relationship, inspired by the work of Berget-Eur-Spink-Tseng. Its consequences include volume polynomial formulae and positivity results for invariants like the interlace polynomial. This talk is based on joint work with Chris Eur, Matt Larson and Hunter Spink.
In this talk we will introduce some notions for positive characteristic rings useful to detect singularities of the corresponding varieties. In particular we will focus on F-singularities of determinantal rings and, using a combinatorial approach, we establish a new upper bound for the F-threshold of rings generated by maximal minors and we compute the exact value in the case of 3×n and 4×n matrices.
Game theory is an area that has historically benefited greatly from outside ideas. In 1950, Nash published a very influential two-page paper proving the existence of Nash equilibria for any finite game. The proof uses an elegant application of the Kakutani fixed-point theorem from the field of topology. This opened a new horizon not only in game theory, but also in areas such as economics, computer science, evolutionary biology, and social sciences. In this talk, we model different notions of equilibria in terms of undirected graphical models.The vertices of the underlying graph of the graphical model represent the players of the game and the dependencies of the choices of the players are depicted with an edge in the graph.This approach brings game theory in contact with the field of algebraic statistics for the first time, which offers a strong foundation for utilizing algebro-geometric tools to solve interesting problems in game theory. This is joint work with Javier Sendra-Arranz and Bernd Sturmfels.
In this talk, I will give a gentle overview of recent developments at the intersection of theoretical particle physics with real, complex, tropical and algebraic geometry -- all held together with deep combinatorics from the theory of matroids and their subdivisions. The beating heart of the construction is the Cachazo-He-Yuan (CHY) integral and its generalization by Cachazo-Early-Guevara-Mizera (CEGM) to moduli spaces of points in higher dimensional projective spaces. The CHY and CEGM integral evaluates to a richly structured rational function which is closely related to the (positive) tropical Grassmannian, a very richly structured object in combinatorial and tropical geometry. I will explain how the notion of "color" in physics motivated Cachazo-Early-Zhang (CEZ) to introduce the chirotopal tropical Grassmannian, which is constructed from realization spaces of oriented matroids other than the that of the positive Grassmannian.
I am going to introduce tropicalization of semi-algebraic sets and discuss applications in real algebraic geometry. In particular, I’ll focus on (pseudo-)moment cones coming from algebraic certificates of positivity. Tropical geometry can illuminate some classical differences between moments and pseudo-moments from a new perspective. This is based on joint work with Greg Blekherman, Felipe Rincon, Cynthia Vinzant, and Josephine Yu.
We study homogeneous polynomials that can be written as sums of (2s)-th powers of degree d forms. Similar to sums of squares these form full-dimensional convex cones for any s,d. The smallest integer k such that any such form (of fixed degree and number of variables) has a length k representation is called the (2s)-Pythagoras number. We show that all even higher Pythagoras numbers tend to infinity for a fixed number of variables (at least three) as the degree increases.
We then study the cone of binary octics that can be written as sums of fourth powers of quadratics to investigate the case of binary forms. This is the smallest case such that we do not consider sums of squares nor powers of linear forms. The 4-Pythagoras number is shown to be 3 or 4 and we also determine the convex structure of this cone.
(joint with Tomasz Kowalczyk)
We study the problem of maximum likelihood (ML) estimation for statistical models defined by reflexive polytopes. Our focus is on the ML degree of these models as a way of measuring the algebraic complexity of the corresponding optimization problem. We compute the ML degrees of all 4319 classes of three-dimensional reflexive polytopes and prove formulas for several general families, which include the hypercube and the cross-polytope in any dimension. We find some surprising behavior in terms of the gaps between ML degrees and degrees of the associated toric varieties, and we encounter some models of ML degree one. This is joint work with Janike Oldekop.
In this talk, we will survey the method of Chabauty and Coleman and its variations. The original method is powerful for determining rational points on curves satisfying a certain rank condition. Depending on this rank condition, we also discuss the other variations that we can use to determine rational (or more general) points. Also, we give a motivation to consider these various cases.
Supergeometry is an extension of geometry to dimensions with anti-commuting coordinates as was motivated by supersymmetry in high-energy physics. Super Riemann surfaces are generalizations of Riemann surfaces with spin structure and have one complex commuting dimension and one anti-commuting dimension. Many aspects of super Riemann surfaces have been investigated and found to mirror and extend classical results on Riemann surfaces in an interesting way.
In this talk, I want to give an overview on super Riemann surfaces and the resulting moduli spaces of stable super curves and stable super maps.
Secant varieties are among the main protagonists in tensor decomposition, whose research covers both pure and applied mathematical areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this talk we discuss the singular locus of the secant variety of lines to a Grassmannian Gr(k,V) using its structure as SL(V)-variety, also solving the problems of identifiability and tangential-identifiability of points in the secant variety. This is based on a joint work with Reynaldo Staffolani.
A matroid is realizable if we can obtain its bases from the indices of linearly independent columns of some matrix. For a given matroid $M$, this matrix is not unique. The space of all such matrices can be given the structure of an affine scheme, known as the realization space of $M$. It is known that representation spaces of matroids can be arbitrarily singular, although there are few concrete examples. We use software to study smoothness and irreducibility of representation spaces of rank 3 and rank 4 matroids, isolating examples of singular spaces for $(3,12)$-matroids. As an application, we show that singular initial degenerations exist for the $(3,12)$-Grassmannian.
We discuss one of the possible generalizations of the spectral theory of matrices to the tensor setting. Motivated by applications in hypergraph theory, we study the characteristic polynomial of a tensor and investigate the problem of recostructing a tensor from its eigenvalues. We focus on the setting of symmetric tensors, highlighting connections to intersection theory and projective geometry. This is based on joint work with Francesco Galuppi, Ettore Teixeira Turatti and Lorenzo Venturello.
Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a filtering function on it, one can in fact consider a filtration given by the sublevel sets of the space induced by the function, and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we exploit such a grid to understand the geometry of a metric between filtering functions and the homological invariants associated with them, called the matching distance. This talk is based on joint work with Marc Ethier, Patrizio Frosini and Nicola Quercioli.
The combinatorial structure of a subspace arrangement can be captured by a polymatroid. The polymatroid arising from the image of the subspace arrangement under a linear map is in an intricate relation with the original polymatroid. This leads to the notion of quotients for submodular functions and M-convex sets.
We lay the foundation for quotients of more general discrete convex functions by giving several equivalent definitions of quotients for M-convex sets. In the talk, I will give a basic introduction to the necessary notions from discrete geometry and matroid theory followed by an overview of new insights.
It is based on joint work with Marie Brandenburg and Ben Smith.
In 1945, Siegel showed that the expected value of the lattice-sums of a function over all the lattices of unit covolume in an n-dimensional real vector space is equal to the integral of the function. In 2012, Venkatesh restricted the lattice-sum function to a collection of lattices that had a cyclic group of symmetries and proved a similar mean value theorem. Using this approach, new lower bounds on the most optimal sphere packing density in n dimensions were established for infinitely many n.
In the talk, we will outline some analogues of Siegel’s mean value theorem over lattices. This approach has modestly improved some of the best known lattice packing bounds in many dimensions. We will speak of some variations and related ideas.
(Joint work with V. Serban, M. Viazovska)
A matroid M is uniformly dense if rank(M|X)/|M|X|≥rank(M)/|M|for all nonempty restrictions M|X. These matroids are extremal for certain connectivity, packing and covering properties and have applications in the design of robust networks. In this talk, I will discuss a new characterization of uniform density derived from the geometry of matroid polytopes and some of its consequences. As a first application, using the inverse moment map we show that uniformly dense real matroids (i.e. real matrices) are parametrized by a subvariety of the Grassmannian. In the case of positroids, this becomes a linear section with the nonnegative Grassmannian. Second, we show that regular uniformly dense graphic matroids have strong connectivity properties and admit a perfect matching. To conclude, I will mention a number of open problems related to uniform density: some polytopes, positroids and a conjecture.
This is joint work with Raffaella Mulas, available on https://arxiv.org/abs/2306.15267.
The inference of phylogenetic networks, essential for understanding evolutionary relationships involving hybridization and horizontal gene transfer, presents formidable challenges in both theory and practice. While standard phylogenetic methods can infer gene trees from genetic data, these trees only indirectly reflect the species network topology due to horizontal inheritance and incomplete lineage sorting.
Previous research has shown that certain network topologies and numerical parameters can be identified, but gaps remain in understanding the full topology of level-1 phylogenetic networks under the Network Multispecies Coalescent model. In this talk, we will aim to fill these gaps and address both, the identifiability of the full topology of the network, as well as the numerical parameters by investigating the ideals defined by quartet concordance factors for topological semi-directed networks.
A hyperplane arrangement is called free if its module of derivations is free. A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices.
In this talk, we first present the concept of freeness in the graphic setting and extend it to the case of graphic arrangements of projective dimension at most 1, whose underlying graphs form the class of weakly chordal graphs (a graph is weakly chordal if the graph and its complement have no chordless cycle with five or more vertices).
A positive geometry is a certain type of space that is equipped with a canonical meromorphic form. While the construction originates in theoretical physics, many beloved objects in algebraic combinatorics and geometry turn out to be examples of positive geometries. In this talk, I will focus on one such example: polytopes. Given any convex polytope, we will study its corresponding “wonderful” polytopes, which arise from the wonderful compactification of a hyperplane arrangement in the same way that polytopes arise as the regions of a hyperplane arrangement. I will describe on-going work with Chris Eur, Lizzie Pratt, and Raluca Vlad showing that any simple wonderful polytope is a positive geometry. I aim to make this talk accessible, and no prior knowledge of positive geometries or wonderful compactifications will be assumed.
In this talk, I will present an efficient approach for counting roots of polynomial systems, where each polynomial is a general linear combination of fixed, prescribed polynomials. Our tools primarily rely on the theory of Khovanskii bases, combined with toric geometry.
I will demonstrate the application of this approach to the problem of counting the number of approximate stationary states for coupled Duffing oscillators. We have derived a Khovanskii basis for the corresponding polynomial system and determined the number of its complex solutions for an arbitrary degree of nonlinearity in the Duffing equation and an arbitrary number of oscillators. This is the joint work with Paul Breiding, Mateusz Michalek, Javier del Pino, and Oded Zilberberg.
Parametric polynomial systems with fixed support that arise in applications often have algebraic dependencies between the coefficients, which makes them more intricate to study than sparse systems where the coefficients are completely free. For example, the generic dimension of the solution set might be higher than the one predicted by the supports and number of equations, and in the zero-dimensional case, the generic cardinality might be lower than the one predicted by Bernstein’s theorem.
In this talk, we will look closer at these issues for the steady state equations studied in chemical reaction network theory. In the first part, I will discuss various network-theoretic conditions that ensure that the codimension of the steady state variety generically is the rank of the network, and that it generically intersects the stoichiometric compatibilities classes finitely. In the second part of the talk, I will discuss a tropical generalization of Bernstein’s theorem that allows us to compute the generic number of complex steady states in a stoichiometric compatibility class, by replacing the mixed volume with a tropical intersection number. This, in turn, also gives us optimal start systems for numerically approximating the steady states homotopy continuation, without tracing superfluous paths, and makes it possible to certify that all of them are found.
This is a combination of several joint works with Elisenda Feliu, Paul Helminck, Beatriz Pascual-Escudero, Yue Ren, Benjamin Schröter, and Máté Telek.
Tropical geometry has a nice connection with toric intersection theory. More precisely, certain intersection numbers that appear in toric intersection theory can be interpreted as multiplicities of certain tropical varieties. In this talk, I will explain this connection more in detail, and show how to get an algorithm out of it that allows us to compute the intersection class of a subvariety of a toric variety from the data of its tropicalization. Finally, I will discuss some applications and open problems on Wonderful Compactifications.
Some of the central objects in string theory are certain integrals on the moduli spaces of punctured Riemann surfaces. In this talk, I will review recent progress in evaluating them, which involves tools from algebraic geometry, combinatorics, and analytic number theory.
Moduli spaces of graphs/tropical curves appear in various areas of maths and physics, for instance in geometric group theory, algebraic geometry/topology, and perturbative quantum field theory. They provide nice venues for combinatorics, algebra, geometry and topology to interact in interesting and fruitful ways. Quite recently, people have started to think about differential forms and integration on these spaces (they are far from being smooth manifolds). In this talk I will focus on two kinds of forms/integrals, Feynman integrals and "canonical integrals of invariant forms" which were recently constructed by Francis Brown. The former case can be thought of as a "1960's version" of the amplituhedron. The latter case provides a de Rham theory for Kontsevich's commutative graph complex. This relies on the fact that these moduli spaces are geometric models for various graph complexes (which in turn relate to various invariants in low-dimensional topology and group theory). I will explain this geometric viewpoint and introduce some interesting subspaces. One particular example is the "spine" of the moduli space of graphs/tropical curves which is a rational classifying space for Out(F_n), the outer automorphism group of a free group. Here it is then natural to ask if there also exist invariant forms on such subspaces and how to find/construct them. I will discuss a (physics-inspired) geometric approach to this problem and explain how this may shed new light on the overall structure of the homology of Out(F_n).
This is a sequence of two understandable lectures, each 50 minutes, with an in-between break for questions.
The speaker will present some material from his excellent course notes "Physics of the Analytic S-Matrix” (2306.05395).
Many members of our Nonlinear Algebra group (including Bernd) have never taken a class in physics. This event is meant for them.
I will review a surprising connection between the scattering of elementary particles in physics and the geometric object called amplituhedron, not known to mathematicians until relatively recently. It is a substantial generalization of the positive Grassmannians, studied in the last two decades in the context of plabic graphs and cluster algebras. The amplituhedron provides an excellent playground for new ideas in mathematics which have a direct impact on solving half-century old questions on all-loop order calculations in quantum field theory, but even more ambitiously can provide a path to a completely new mathematical framework for the description of fundamental laws of Nature.
In this talk, we study a hypergeometric integral associated with any Laurent polynomial. It is called Feynman integral in physics and called marginal likelihood integral in statistics. A twisted cohomology group is a system of difference equations that hypergeometric integral satisfies. This is a left ideal in a non-commutative ring. It naturally "converges to" likelihood equations previously studied in algebraic statistics. The converse operation exists in principle: the likelihood equation knows the twisted cohomology. We will clarify the meaning of this statement. Based on joint work with Simon Telen (MPI MiS) arXiv:2301.13579.
A chopped ideal is obtained from a homogeneous ideal by considering only the generators in a (low) degree. When the original ideal defines a sufficiently small number of points in projective space, chopping it does not alter the scheme. The complexity of computing these points from the chopped ideal is governed by the Hilbert function. We conjecture the values of this function and prove it in several cases. Using symbolic methods, we verify the conjecture for a large range of points. Our study of chopped ideals is motivated by symmetric tensor decomposition.
Suppose we are given a multivariate polynomial over the integers. What can be said about the zeros of this polynomial modulo a natural number N? As I will explain, considering this question for all natural numbers simultaneously leads to a data set with remarkable structure and symmetry.
One route to appreciate this hidden information goes via p-adic integration. I will illustrate this approach -- hands-on and from scratch -- on a few examples.
I will also try and convince you that p-adic integrals are the tool of choice to tackle a number of other counting problems, seemingly less algebro-geometric, say of group-theoretic origin.
I will assume nothing more from my audience than what it takes to understand this abstract's first sentence.
Integer programming (IP) represents a fundamental problem in discrete optimization, and it is of both high theoretical and practical importance. However, solving instances of integer programming is in general NP-hard, and therefore, a long line of research has been devoted to identifying classes of IP that can be solved efficiently. One of the most prominent cases is IP in fixed dimension, which can be solved in polynomial time due to the famous result of Lenstra. As for algorithms for IP in variable dimension, recent developments show that IP can be solved efficiently when the constraint matrix admits a block-like structure -- this is represented for example by the classes of n-fold, 2-stage stochastic, or multi-stage stochastic integer programming. A large family of algorithms for solving IP in variable dimension has been based on the framework of iterative augmentation. The idea of this framework is similar to computing the maximum flow on graphs -- starting with an initial feasible solution to IP, we iteratively apply improving steps until we converge to an optimal solution. The notion of Graver basis then represents the set of improving steps. However, the Graver basis of the matrix can be very large, and to avoid explicit computation, the algorithms rely on the bounds on the ℓ1 or ℓ∞-norm of the elements of the Graver basis.
In this talk, we describe the connection between the norms of the elements of the Graver basis and matroid depth parameters. In particular, we give a structural characterization of matrices with small ℓ1 -norm, and as a corollary show that IP is fixed-parameter tractable for parameterization with this notion. Moreover, we briefly discuss the connection of matroid theory to linear programming and the circuit imbalance measures.
The talk is based on the results of joint work with M. Briański, M. Koutecký, D. Kráľ, and F. Schröder.
We study Gibbs varieties associated to ensembles of Hamiltonians obtained from a graph. This results in the notion of a quantum graphical model, providing a generalisation of classical graphical models studied in algebraic statistics.
This is ongoing work with Eliana Duarte and Dmitrii Pavlov.
We use Newton polytopes to construct prime modules of quantum affine algebras, and we apply prime modules to Grassmannian string integrals in physics.
This is joint work with Nick Early.
We study the structure of all possible affine hyperplane sections of a convex polytope, and we craft algorithms that compute optimal sections for various combinatorial and metric criteria.
This is joint work with Jesus De Loera and Chiara Meroni.
We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of L shaped polygons where the algebraic curve is already known. The algorithm is then implemented for a family of translation surfaces called Jenkins--Strebel representatives that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.
Lorentzian polynomials, recently introduced by Brändén and Huh, have coefficients that satisfy a form of log-concavity, and have been used to prove, reprove, and conjecture various combinatorial statements coming from convex geometry, representation theory, and the theory of matroids. Accepting that being Lorentzian is a useful concept, it is natural to ask for differential operators (with constant coefficients) that preserve this property. This leads to the notion of dually Lorentzian polynomials. As an application of this observation, I will show how dually Lorentzian polynomials give rise to generalisations of the Alexandrov-Fenchel inequality in convex geometry.
This is joint work with Julius Ross and Thomas Wannerer.
This is an informal introductory lecture on local cohomology. We will follow Appendix A of Eisenbud's book 'The geometry of syzygies'. Participants are encouraged to have a look at this text beforehand.
The Euclidean Distance degree EDD(Q,X) of an algebraic variety X in a real inner product space (V, Q) counts the number of complex critical points of the distance function from a generic point in V to X. Since this invariant of X depends on Q, it is a natural problem to find or characterize inner products Q that correspond to the minimal possible EDD(Q,X). In my talk I will discuss this question for Segre-Veronese varieties, which consist of rank-1 (partially symmetric) tensors. I will show that with respect to the classical Frobenius (a.k.a. trace) inner product F(A,B)=Tr(AB), the variety X of nxm rank-1 matrices has smallest EDD(F,X)=min(n,m), whereas EDD(Q,X) with respect to a sufficiently general inner product Q on the space of nxm matrices is much higher.
Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of the singular locus of the hypersurface associated to the tensor. In this talk, we will first learn fundamental properties of the symmetric geometric rank. Then, we will study the space of symmetric tensors of prescribed symmetric geometric rank, which is the space of homogeneous polynomials whose corresponding hypersurfaces have a singular locus of bounded codimension.
This is joint work with J. Lindberg.
Many processes in natural sciences can be described by dynamical systems in the state-space form or, alternatively, by algebraic input-output equations. The question of recovering a dynamical system from a given input-output equation is known as the realizability problem. In this talk we give an overview of algebraic approaches to this problem when the dynamical system is restricted to have rational right-hand sides, and concentrate on finding realizations of input-output equations over the field of real numbers.
This short talk is an advertisement for the Summer School which will be run by Alicia Dickenstein, Elisenda Feliu and Timo de Wolff in June at MPI Leipzig. Teaser: What is a "toric dynamical system” ?
Given a 2n-dimensional vector space V with a symplectic form w, its linear subspace L is called isotropic if all vectors in L are pairwise orthogonal with respect to the form w. The symplectic Grassmannian SpGr(k,2n) is the space of all k-dimensional isotropic linear subspaces of V. We formulate tropical analogues of several equivalent characterizations of this space. These tropical analogues are not equivalent in general; we give all implications between them, and some counter examples. In the case k=2, we study the fan structure of the respective tropical symplectic Grassmannian, giving a count of the number of rays and maximal cones.
This is based on joint work with Jorge Alberto Olarte.
Arkani-Hamed, Benincasa and Postnikov defined a cosmological polytope associated to a Feynman diagram in their study of the wavefunctions associated to certain cosmological models. By computing the canonical form of this polytope, one computes the contribution of the Feynman diagram to the wavefunction of interest. The theory of positive geometries tells us that one way to compute this canonical form is as a sum of the canonical forms of the facets of a subdivision of the polytope. For simple examples of Feynman diagrams, it is known that specific triangulations correspond to classical physical theories for these computations, but a general theory of triangulations of cosmological polytopes was left as future work. In this talk, we will discuss an algebraic approach to this theory based on Gröbner bases. We will see that every cosmological polytope admits families of regular unimodular triangulations whose facets can be understood graph theoretically. We characterize the facets of these triangulations for certain families of Feynman diagrams, including trees and cycles. In addition to providing possible new physical theories for the computations of the associated wavefunctions, these results also allow us to recover combinatorial information about these polytopes, such as normalized volumes, extending some recent results of Kühne and Monin.
This talk is based on joint work with Martina Juhnke-Kubitzke and Lorenzo Venturello.
Given a smooth complex manifold, Chern-Weil theory asks for the Chern classes of holomorphic vector bundles to be represented by forms and currents in de Rham cohomology. If the vector bundle is endowed with a smooth (even mildly singular) metric, then Chern Weil theory holds. There are however cases of rich arithmetic interest (e.g. universal abelian varieties) where Chern-Weil theory does not longer hold. The goal of this talk is to propose an infinite version of Chern-Weil theory for line bundles using tropical geometry. The key idea is that the algebraic geometric analogue of the first Chern current of a singular metric should actually be a limit of divisors, a so-called b-divisor. Assuming some toroidal properties of the metric, this limit can be expressed in terms of a function on a tropical variety. Then, top wedge products of Chern currents of singular metrics correspond to a mixture of algebraic and tropical intersection numbers.
Scattering amplitudes are one the most fundamental observables in physics and in recent years it has been appreciated how they are intimately connected with various branches of discrete mathematics, such as Combinatorics and Tropical Geometry. The recurrent theme of these connections is that the singularity properties of scattering amplitudes, such as the patterns of poles and residues that they are allowed to have, are mirrored by the boundary structure of certain geometrical objects defined by some notion of positivity.
In this talk I will review a recent and particularly simple instance of this general phenomenon which sees on the physical side of the correspondence a colored cubic scalar field theory, and on the mathematical side a class of convex polytopes that describe the combinatorics of the crossing of curves on Riemann surfaces. The emphasis will be put on describing these new polytopes, dubbed Surfacehedra, which generalize the classical Associahedron to any surface.
The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define y-rational Gorenstein polytopes, which extend the classical notion to the rational setting. This is joint work with Matthias Beck and Sophia Elia.
I will talk about some aspects of the latest research that we (Yuri Ivanovitch and I) have been doing since 2018 until January 7, 2023. This includes advances in Frobenius Manifolds, Grothendieck—Teichmuller theory, Information theory and their hidden symmetries.
In this talk we consider the geometry of the image of the even Vandermonde map $\mathbb{R}^n$ to $\mathbb{R}^d$ consisting of the first d even power sums. For fixed degree d, the images form an increasing chain in the number of variables. We give a description of the image for finite n and at infinity, and prove that the image has the combinatorial structure of a cyclic polytope.We show how the image of the Vandermonde map relates to the study of copositive symmetric forms and prove undecidability of verifying nonnegativity of trace polynomials whose domains are all symmetric matrices of all sizes.This is joint work with Jose Acevedo, Greg Blekherman and Cordian Riener.
We sketch the proofs of the closure properties of D-algebraic functions involved in the duplication formula of the Weierstrass elliptic function. In particular, we show that this formula can be automatically verified.
This is a glimpse of joint work with Rida Ait El Manssour and Anna-Laura Sattelberger.
Homological Mirror Symmetry (HMS) is a conjecture (proven in some cases) relating the A-model of a manifold with the B-model of its mirror dual manifold; the A-model comprises symplectic geometry whereas the B-model is complex-algebraic. More precisely, the A-model is given by a Fukaya category and the B-model is given by the derived category of coherent sheaves; HMS establishes an equivalence between these two categories.In this talk we will introduce the concept of Fukaya categories and present a calculation of the wrapped Fukaya category of $\mathbb{C}^*$, thereby proving that $\mathbb{C}^*$ is mirror dual to itself. Moreover, partially wrapped Fukaya categories and their combinatorial descriptions as marked surfaces will be introduced, stating HMS for $\mathbb{A}_{\mathbb{C}}^1$ and $\mathbb{P}_{\mathbb{C}}^1$. In the end, we will point out connections to matrix factorizations appearing in the B-model of the Landau—Ginzburg model mirror dual to $\mathbb{P}_{\mathbb{C}}^1$.
The essential variety is an algebraic subvariety of dimension $5$ in $\mathbb RP^3.$ It encodes the relative pose of two calibrated cameras, where a calibrated camera is a matrix of the form $[R,t]$ with $R\in SO(3)$ and $t\in \mathbb R^3$. Since the degree of this variety is $10$, there can only be at most $10$ complex solutions. We compute the expected number of real points in the intersection of the essential variety with a random linear space of codimension $5$. My aim is to tell you about these computations and our results. This is joint work with Paul Breiding, Samantha Fairchild, and Pierpaola Santarsiero.
In this talk, we will introduce some ideas for the tropicalization of varieties with respect to valuations of higher rank. We will show that these spaces have the structure of iterated fibrations of usual tropical varieties, and we will explain how to understand these fibrations in the case of hypersurfaces. Moreover, we will outline a theory of polyhedral geometry over the ordered ring of real numbers extended with a nilpotent infinitesimal R[x]/(x^n), which will allow us to endow higher rank tropical varieties with the structure of a polyhedral complex over this ring.
Tropical geometry studies degenerations of algebraic varieties by enriching the theory of semistable models and their dual complexes by polyhedral geometry. This enrichment motivates the development of algebraic geometry for combinatorial and polyhedral spaces.
While the theory has been largely developed over the past two decades and has found diverse applications, the framework has been mostly restricted so far to the case of valuations of rank one. From the geometric point of view, this means considering families of complex manifolds which depend only on one parameter. There are several reasons for a desire to extend the scope of tropical geometry beyond the rank one case. For example, one of the leitmotifs in the development of tropical geometry is to explain large scale limits of complex geometry. From the point of view of moduli spaces and their compactifications, the problem should be understood using higher rank valuation theory since large scale limits can depend on several parameters.
In this talk, we present some interesting features of valuation theory in the higher rank setting. In particular, we introduce an appropriate higher rank notion of dual complexes, and discuss some applications to geometry, in the study of Newton-Okounkov bodies and in asymptotic complex geometry, making a connection to the work of the speaker with Nicolussi.
Based on joint work with Hernan Iriarte.
One can model a data set as a metric space or a metric measure space with a function, which I am going to call a field. For example, a weighted network with labeled vertices can be modeled as a metric measure space with a function by endowing the set of vertices with the shortest path distance and normalized counting measure. As data sets are noisy, constructions applied to them should be stable, in the sense that similar data sets should produce similar outputs. This requires a method to measure the degree of similarity between the objects used for modeling data sets, in particular fields.In this talk, I am going to introduce analogues of Gromov-Hausdorff and Gromov-Wasserstein distances for fields, and state some of their properties. Then, I will obtain a geometric representation of isomorphism classes of fields under these metrics through the construction of Urysohnn universal fields.(Joint work with S. Anbouhi and W. Mio.)
In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this presentation, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. More precisely, we show that the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called infectivity (=hyperconvexity). This permits proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants, for example the notion of spread introduced by M. Katz. As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M. Katz, (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F. Wilhelm, and (3) some nontrivial lower bounds of Gromov-Hausdorff distance between model spaces via the stability lemma.
We consider two topological transforms that are popular in applied topology: the Persistent Homology Transform (PHT) and the Euler Characteristic Transform (ECT). Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset $M$ of $ℝ^d$, and associates to each direction $v ∈$ $S^{d−1}$ a shape summary obtained by scanning $M$ in the direction $v$. These shape summaries are either persistence diagrams or piecewise constant integer-valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes, i.e. each shape has a unique transform. Moreover, we prove that these transforms determine continuous maps from the sphere to the space of persistence diagrams, equipped with any Wasserstein $p$-distance, or the space of Euler curves, equipped with certain $L^p$ norms. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable space of PL embedded shapes with plausible geometric bounds can be uniquely determined using only finitely many directions. This provides a complexity metric for a moduli space of shapes.
Learning with neural networks relies on the complexity of the representable functions but, more importantly, the particular assignment of typical parameters to functions of different complexity. Taking the number of activation regions as a complexity measure, we show that the practical complexity of networks with maxout activation functions, which correspond to tropical rational maps, is often far from the theoretical maximum. Continuing the analysis of the expected behavior, we study the expected gradients of a maxout network with respect to inputs and parameters and obtain bounds for the moments depending on the architecture and the parameter distribution. Based on this, we formulate parameter initialization strategies that avoid vanishing and exploding gradients in wide networks.
In 1976 M. Atiyah, in connection with his work with Singer, introduced a certain kind of cohomology in order to extend his famous Atiyah-Singer Index Theorem to the noncompact setting. This cohomology is nowadays commonly known as l2-cohomology. He also defined the generalized notion of l2-Betti numbers as von Neumann dimensions of the resulting l2-cohomology groups. In his paper Atiyah observed that in many cases these are in fact rational numbers, and indeed he asked if it is possible to obtain irrational ones. That was the beginning of what is now called the Atiyah conjecture.
In this talk I will introduce the basic notions of von Neumann dimension and l2-Betti numbers in a purely algebraic setting. I will also present some modern variants of the Atiyah conjecture, as well as known results concerning them.
Lastly, I will talk about other famous conjectures on group rings, and their relations with the previously stated Atiyah conjecture.
Given a compact, convex set $K\subset \mathbb{R}^{n+m}$. Consider the projection of $K$ to the first $n$ coordinates. Over every point in $\mathbb{R}^n$ we have a convex fiber. The fiber body is "the average" over all such fibers. Gram spectrahedra are precisely the fibers of a linear map where $K$ is the cone of positive semidefinite matrices. Therefore, the fiber body of Gram spectrahedra is "the average" Gram spectrahedron in some sense. We study the boundary of fiber bodies for Gram spectrahedra of binary/ternary forms of low degree. (This is joint work in progress with Chiara Meroni.)
We study combinatorial and geometric properties of toric degenerations of Cox rings of blow-ups of $\mathbb{P}^3$ at points in general positions. We focus in particular on Ehrarht-type formulas for the multigraded Hilbert functions of these spaces. From our computations, it follows that the presentation ideal of the Cox ring of the blow-up of $P^3$ at seven points is quadratically generated, as conjectured by Lesieutre and Park. The talk is based on recent work with Mara Belotti.
Finding rational solutions of first-order algebraic ordinary differential equations with functional coefficients is in general a hard problem. By considering these coefficients as parameters, we obtain an implicitly defined curve. In this talk, we study rational parametrizations of such algebraic curves involving unknown parameters leading to a procedure for finding rational solutions of the original differential equation.
In algebraic geometry and number theory, objects often naturally come equipped with a monodromy or Galois action. Studying this action gives us insight into the structure of the object. Harris initiated the study of the monodromy of enumerative problems, like "how many lines are on a cubic surface?''. A Fano problem is an enumerative problem of counting linear subspaces on complete intersections in projective space, like counting lines on a cubic surface or 2-planes on the intersection of 3 quadrics in $P^8$. In this talk, I discuss the monodromy of Fano problems, and a proof that the monodromy groups of most Fano problems are large. This is joint work with Borys Kadets.
The approximate vanishing ideal of a set of points $X\subseteq \R^n$ is the set of polynomials that approximately evaluate to $0$ over all points $\textbf{x} \in X$ and admits an efficient representation by a finite set of polynomials called generators. The constructed generators capture polynomial structures in data and give rise to a feature map that can, for example, be used in combination with a linear classifier for supervised learning. Generator-constructing algorithms are widely studied, but their computational complexities remain expensive and the methods lack learning guarantees. We introduce a generator-constructing algorithm that admits several learning guarantees and whose computational complexity is linear in the number of samples.
Relative entropy programs belong to the class of convex optimization problems. Within techniques based on the arithmetic-geometric mean inequality, they facilitate to compute nonnegativity certificates of polynomials and of signomials (i.e., exponential sums). While the initial focus was mostly on unconstrained certificates and unconstrained optimization, recently, Murray, Chandrasekaran and Wierman developed conditional techniques, which provide a natural extension to the case of convex constrained sets. The goal of this talk is to explain the geometry of the resulting cone ("conditional SAGE cone"). To this end, we introduce and study the sublinear circuits of a finite point set in R^n, which generalize the simplicial circuits of the affine-linear matroid induced by a set A to a constrained setting. Based on joint work with R. Murray and H. Naumann.
There are several ways of computing the degree of varieties, but the most basic one is to intersect your variety with a line. We will be doing the same in the tropical world. With the techniques developed we will compute the likelihood degeneracy degree of a matroid and recover a result from Agostini et al. In this talk, we will introduce the Bergman fan of a matroid, our tropical variety of interest, study its combinatorics, and identify its degree with the number of so called nbc bases.
Matroids are central players in modern combinatorial algebraic geometry. One algebro-geometric way to find a matroid is by taking any point in the Grassmannian. Then there is an associated matroid that corresponds to the set of Plücker coordinates of that point which are nonzero. Furthermore, for a certain canonical torus action, the moment map of the orbit of any point in the Grassmannian yields a polytope associated with the matroid of the point called the matroid base polytope. We will discuss a similar story for orbits of points in full flag varieties (i.e. flag varieties for flags of subspaces of dimension 0, 1, 2, ..., k of a space of dimension n). Then I will describe my recent work investigating the combinatorics of their moment polytopes and showing that the corresponding toric varieties are smooth. Based on joint work with Raman Sanyal.
The study of Feynman integrals is central for computing observable quantities in high-energy physics. However, many analytic properties of Feynman integrals are still poorly understood in full generality. A key step towards uncovering analytic properties of Feynman integrals is to first understand the space of functions to which they belong. It turns out that Feynman integrals arise as special cases of Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric functions - a class of functions with rich connections to toric geometry, D-module theory, twisted cohomology, and more. In this talk, we shall employ the GKZ framework to obtain Pfaffian systems for Feynman integrals, i.e. a system of 1st-order PDEs obeyed by a vector space basis of integrals. Our algorithm for obtaining Pfaffian systems is based on Macaulay matrices, offering an efficient alternative to the traditional method based on Gröbner bases.
Persistent homology is commonly encoded by functors with values in the category of vector spaces and indexed by a poset. These functors are called tame or persistence modules and capture the life-span of homological features in a dataset. Every poset can be used to index a persistence module, however some of them are particularly well suited. We introduce a new construction called realisation, which transforms posets into posets. Intuitively it associates a continuous structure to a locally discrete poset by filling in empty spaces. Realisations share several properties with upper semilattices. They behave similarly with respect to certain notions of dimension for posets that we introduce. Moreover, as indexing posets of persistence modules, they both have good discretisations and allow effective computation of homological invariants via Koszul complexes. This talk is based on a joint work with Wojciech Chachólski and Alvin Jin.
A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids. I will discuss generalized matroid representations as arrangements over division rings, subspace arrangements and as entropy functions together with their relation to group theory. As an application this yields a proof that the conditional independence implication problem from information theory is undecidable. Based on joint work with Rudi Pendavingh and Geva Yashfe.
Is there an intuitive and prerequisite-minimizing way to think about scattering processes in Quantum Field Theory? In this talk, we will explore how far we can go with only the simplest of criteria, seemingly unrelated: octahedra should be subdivided at most once, into two square pyramids.
The boundary of a convex hull is a subtle object. However, there is a technique to study the convex hull of a real variety. The goal is to understand which varieties contribute to the boundary. We analyze in details the case of smooth surfaces in four-dimensional space, in particular for Veronese, Del Pezzo, and Bordiga surfaces.
In this hour, I will present two short lectures, each about 25 minutes long. They feature articles posted recently on the arXiv. The first discusses "Subspaces Fixed by a Nilpotent Matrix", and is coauthored with Marvin Hahn, Gabriele Nebe and Mima Stanojkovski. The second concerns the "Recovery of Plane Curves from Branch Points", where the collaborators are Daniele Agostini, Hannah Markwig, Clemens Nollau, Victoria Schleis and Javier Sendra.
Given a real algebraic curve of degree d in projective space, we may ask whether there exists a hyperplane that meets it in d real points. We will look at this problem and some variations from two different angles: (1) As a computational decision problem. (2) As part of a question about the real divisor class group of the curve. (Based on joint work with Huu Phuoc Le and Dimitri Manevich; arxiv:2106.13990)
We give a method for sampling points from an algebraic manifold (affine or projective) over a local field with a prescribed probability distribution. As an application, we sample from algebraic p-adic matrix groups and modular curves.
Plucker coordinates provide a concrete and useful way to understand the Grassmannians $Gr(k,n)$ parametrizing k-dimensional subspaces of an n-dimensional vector space. In this talk, we will explore Plucker coordinates for more general homogeneous spaces, and for certain homogeneous spaces, give a representation-theoretic computation to find a family of valuations for which the Plucker coordinates form a Khovanskii basis, and hence correspond to lattice points of a Newton-Okounkov body. This is joint work in progress with Peter Spacek.
The topic of nonnegative polynomials on varieties has attracted a lot of modern interest because of their connections to optimization and real algebraic geometry. Stanley-Reisner varieties are simple varieties that are unions of coordinate planes. We will discuss the topic of nonnegative quadratic forms over Stanley-Reisner varieties and how we can classify extreme such quadratic forms. This topic is directly related to questions of positive semidefinite matrix completion and sparse semidefinite programming. We hope to explore some interesting connections between these quadratic forms and the geometry of certain associated simplicial complexes.
Elliptic pairs consist of a surface X and a curve C on X satisfying properties similar to an elliptic curve. They are a useful tool for understanding the cone of effective divisors of X, and interesting geometric objects in their own right. In this talk we will classify elliptic pairs where the surface X is toric and comes from a triangle. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of the projective plane at nine points on a nodal cubic, over a finite field. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes of positive density.
The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish.
Based on Mikhalkin's Correspondence Theorem, tropical geometry has been successfully used to count plane curves satisfying point conditions. For curves there exist nice tools using tropical geometry to break the task of counting curves down to a combinatorial problem. The generalization for higher dimensional varieties, like surfaces, is more involved. After a brief introduction to tropical geometry, I will present tropical floor plans as developed by Markwig et. al. as a tool, which allows to count not only curves but also multi-nodal surfaces under certain constraints. In joint work with Madeline Brandt, we extended the definition of tropical floor plans to count even more cases of surfaces: In particular in our newest paper, we looked at the case when two nodes are tropically close together, i.e., unseparated. We then prove that for δ=2 or 3 nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second order term of the polynomial giving the degree of the family of complex projective surfaces in ℙ3 of degree d with δ nodes. We classified the cases when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and proved that this only happens for projective degree d surfaces satisfying point conditions in Mikhalkin position when d>4.
The use of machine learning in pure mathematics to help formulate conjectures has been a growing research area, with examples in knot theory and representation theory. In this talk, we go through two successful example applications of machine learning algorithms to problems in algebraic geometry where toric varieties are the central object of study. In the first, we study the quantum period of toric varieties, and see if we can learn their dimension from it. In the second, we construct a neural network that distinguishes between terminal and non-terminal toric varieties from their GIT data, which motivates a proposed mathematical solution to the problem.
A statistical model is identifiable if the map parameterizing the model is injective. This means that the parameters producing a probability distribution in the model can be uniquely determined from the distribution itself which is a critical property for meaningful data analysis. In this talk I'll discuss a new strategy for proving that discrete parameters are identifiable that uses algebraic matroids associated to statistical models. This technique allows us to avoid elimination and is also parallelizable. If time permits I'll also discuss a new extension of this technique which utilizes oriented matroids to prove identifiability results that the original matroid technique is unable to obtain.
The story of linkage in codimension two is well-understood, and was originally used to classify curves in $P^3$. For example, such a curve is in the linkage class of a complete intersection (licci) if and only if it is arithmetically Cohen-Macaulay. In codimension three (and beyond) only the forward implication remains true in general, but it is conjectured that if the Betti numbers are "small" then the reverse direction holds. I will share some ongoing joint work with Lorenzo Guerrieri and Jerzy Weyman regarding this topic, with an emphasis on examples. No prior knowledge of linkage is assumed.
We continue our exploration of how to encode mathematical objects and about what this entails for mathematical research. This time we look into Gröbner bases, why they matter and how large they can be. The theoretical analysis in the presentation is complemented by subsequent empirical studies by the participants.
The Borsuk--Ulam theorem has found numerous applications across mathematical disciplines since its discovery in the 1930s. The theorem states that any continuous map from a d-sphere to d-space identifies a pair of antipodal points. I will show that this result remains relevant today and present new consequences. I will present two recent generalizations of the Borsuk--Ulam theorem, a colorful extension and a version for high-dimensional codomains, and explain some connections with packings of projective space among other topics. This is joint work with Henry Adams, Johnathan Bush, and Zoe Wellner.
Two algebraic varieties (or manifolds) are said to be derived equivalent if, roughly, they have the same set of functions on them. We discuss when two genus 1 curves (or genus 1 Riemann surfaces, if you prefer) are derived equivalent. This turns out to be related to when one of the curves is the classifying space of line bundles on the other curve. Background in moduli spaces and line bundles/Picard groups will not be assumed--I'll explain all the words in the abstract. Joint work with Soumya Sankar.
The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. Using these results, we present a homotopy method to perform parameter recovery, and therefore density estimation, for high dimensional Gaussian mixture models. The number of paths tracked in our method scales linearly in the dimension.
Exponential families on a finite set are well-studied in statistics, but they are still a source of many problems. For example, marginal likelihood integral in Bayesian inference for conjugate prior naturally gives rise to a hypergeometric integral whose exact formula/evaluation is unknown. Bayesian inference corresponds to contiguity relation, which is an automorphism of de Rham cohomology group defined by the integrand. When the exponential family is generic, the integral is a Gel'fand-Kapranov-Zelevinsky hypergeometric function and the contiguity structure is described in terms of Pfaffian system (system of 1st order linear P.D.E.'s). It is natural to ask the following question: can we restrict Pfaffian system of a generic exponential family to a non-generic one? In this talk, we propose a method to keep track of such restrictions using a technique of singular boundary value problem. In the latter half of the talk, we pay a special attention to two-way (incomplete) contingency table, a class of exponential family. Restriction method combined with the theory of hyperplane arrangement provides a new combinatorial description of the Pfaffian system. This Pfaffian system generalizes familiar hypergeometric systems such as pFp-1, Appell-Lauricella's FA,FB,FD, Tsuda and Aomoto Gel'fand system. The talk is partially based on an on-going joint work with Vsevolod Chestnov, Federico Gasparotto, Manoj K. Mandal, Pierpaolo Mastrolia, Henrik J. Munch and Nobuki Takayama.
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of D-modules. We present an overview and discuss relations between these approaches. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.
Using data from protein or DNA sequences, evolutionary biologists seek the true ancestral relations among species which is represented in the form of a tree. The space of phylogenetric trees is a tropical linear space which motivates a tropical approach to analyze evolutionary data.In this talk, we will focus on the Fermat--Weber problem under an asymmetric tropical distance, emphasizing its geometric and computational aspects. In particular, we will show its connection to the transportation problem from operations research. These properties lead to an efficient consensus method for phylogenetics.The talk is based on joint work with Michael Joswig.
In Polynomial Optimization, finite convergence of the Lasserre's Moment and Sums of Squares hierarchies is usually observed in applications, but it is not completely investigated theoretically. In practice, finite convergence is certified using Flat Truncation, a rank condition on the moment matrix of the sequence of moments that realize the minimum. We investigate the Flat Truncation property, studying Lasserre's spectrahedral outer approximations of the convex cone of measures supported on a semialgebraic set. We present different pathological examples and introduce a new generic algebraic condition that is necessary and sufficient for Flat Truncation. Finally, we deduce convergence rates for Lasserre's spectrahedral outer approximations to the cone of measures from a new version of the Effective Putinar's Positivstellensatz. Based on joint works with Bernard Mourrain and Adam Parusinski.
In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a rational point, and finite dimensional vector spaces of differential operators with constant coefficients. The question for general ideals was left open. Significant progress was made in the 1960's by analysts, culminating in a deep result known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and modules to solution sets of linear, homogeneous partial differential equations with constant coefficients. This talk aims to survey classical results, and provide recent constructions, applications, and insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the analysis of PDEs. This framework is amenable to the development of symbolic and numerical algorithms. We also study some applications of algebraic methods in problems from analysis.
Persistent homology is one of the key concepts in topological data analysis and an active area of research in computational topology. It associates to a filtered simplicial complex the system of the homology vector spaces of each complex in the filtration. This collection can be viewed as a k[x]-module, and a common goal is to compute a free presentation of this module, which can be succinctly described by the so-called barcode. A common optimisation scheme in current software exploits the fact that the computation of persistent cohomology, albeit yielding equivalent results, can be carried out far more efficiently. Analogously, the system of homology vector spaces of a two-parameter filtration can be viewed as a k[x, y]-module. Computing a free presentation of it is more involved, though, and the efficiency of existing implementations lags behind that of one-parameter persistent homology software. This is because optimisations using cohomology cannot be applied straightforwardly anymore, due to the fact that, unlike the one-parameter case, cochain modules are not free anymore.I will show how cohomology can be used to develop efficient algorithms for two-parameter persistence nevertheless by considering free resolutions of cochain modules instead, using a result that links free resolutions of persistent homology and cohomology.
Geometric tomography is concerned with reconstructing shapes from geometric data such as volumes of sections and support function evaluations, a task that arises naturally in a variety of application areas, for example, robotics, computerized tomography and magnetic resonance imaging. In this talk we consider the task of reconstructing polytopes with fixed facet directions from finitely many (possibly noisy) support function evaluations. For fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction. We show that under mild assumptions the least-squares estimate converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed. This is joint work with Maria Dostert.
For a fixed integer d, how many number fields are there of fixed degree d of absolute discriminant less than X? Denote this number N(d, X). This talk will be a friendly foray through the known results, proofs, and main conjectures regarding this question. In particular, it is known that if d = 2,3,4,5 then N(d, X) = O(X). We will discuss the key components of the proofs, which involve the parametrizations of number fields and geometry of numbers. For n >= 6, we will discuss Malle's conjecture, which predicts the asymptotics of N(d, X) for general degree d. We will also discuss classical upper bounds on N(d, X).
Toric varieties have a strong combinatorial flavor: those algebraic varieties are described in terms of a fan. Based on joint work with M. Borinsky, B. Sturmfels, and S. Telen, I explain how to understand toric varieties as probability spaces. Bayesian integrals for discrete statistical models that are parameterized by a toric variety can be computed by a tropical sampling method. Our methods are motivated by the study of Feynman integrals and positive geometries in particle physics.
Mixed areas and mixed volumes are central objects in the measure-theoretic convexity theory. They also play a fundamental role for applications in stochastic, algebraic and tropical geometry, with the link to algebraic and tropical geometry established via the celebrated BKK theorem. Still, some very basic questions on the relations between mixed volumes within a collection of convex bodies remain unsolved. For example, already the exact relations of the 10 mixed areas $V(K_i,K_j)$, $1 \le i \le j \le 4$, of four planar convex bodies $K_1,...,K_4$ are not known and it is not even know if the exact relations are semi-algebraic. Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines a vector of ${n\choose 2}$ pure mixed areas $V(K_i,K_j)$ for $1\leq i
We discuss new necessary and sufficient conditions for the entire functions with positive Taylor coefficients to belong to the Laguerre--Pólya class. It is an important class of entire functions which are locally the limit of a sequence of real polynomials having only real zeros. For an entire function $f(z) = \sum_{k=0}^{\infty} a_k z^k,$ we define the second quotients of Taylor coefficients as $q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_{n}}, n\geq 2$ and find conditions on $q_n(f)$ for $f$ to belong to the Laguerre--Pólya class. Besides, we show the relation of the conditions to the partial theta function $g_a =\sum _{k=0}^{\infty} \frac {z^k}{a^{k^2}}$, when $a>1.$
Invariant Theory is a rich branch of algebra that originated from a work by Cayley in 1845. It has lead to outstanding results such as Hilbert’s celebrated papers in the 1890’s or Mumford’s development of Geometric Invariant Theory (GIT) for reductive groups. In recent years there has been great interest in computational aspects of Invariant Theory and its applications, and at the theoretical frontier non-reductive GIT is being developed. The goal of this talk is, first, to advertise Invariant Theory and, second, to explain why reductive GIT „works well“ - thereby giving some insight which challenges non-reductive GIT needed to overcome. For this, we treat the following topics: reductive groups, stability notions and their applications, the Hilbert-Mumford criterion and the Kempf-Ness theorem. At the end, we catch a glimpse of non-reductive GIT.
Balls in the tropical metric are a particular example of polytropes, namely tropical polytopes that are at the same time convex in the usual sense. Moreover, tropical balls can also be seen as one-apartment slices of balls in Bruhat-Tits buildings. So balls in this context have vertices and we will see how these vertices play an important role in applications to representation theory and algebraic coding theory. I will be reporting on joint work with Yassine El Maazouz, Marvin Hahn, Gabriele Nebe, and Bernd Sturmfels.
It has been known since at least the time of Poincaré that isometries of 3-dimensional hyperbolic space $ \mathbb{H}^3 $ can be represented by $ 2 \times 2$ matrices over the complex numbers: the matrices represent fractional linear transformations on the sphere at infinity, and hyperbolic space is rigid enough that every hyperbolic motion is determined by such an action at infinity. A discrete subgroup of $ \PSL(2,\mathbb{C}) $ is called a Kleinian group; the quotient of $ \mathbb{H}^3 $ by the action of such a group is an orbifold; and its boundary at infinity is a Riemann surface. The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in $ \mathbb{C} $, and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own (joint with Gaven Martin and Jeroen Schillewaert) which extend the work of Keen and Series.
The Ingleton inequality is a necessary condition for a matroid to be linearly representable and it comes in the form of a linear inequality in its rank function. In a probability-theoretic reinterpretation of the inequality, linear subspaces are replaced by discrete random variables and ranks by Shannon entropies. In this setting, the Ingleton inequality no longer holds universally for representable rank functions but only if additional linear constraints are assumed. In this talk, I give an overview of Milan Studený's recent systematic work on these so-called conditional Ingleton inequalities, their historical roots and my own contribution to finishing their classification for four discrete random variables.
In classical combinatorics, matroids generalize the notion of linear independence of vectors over a field. In this talk, we will introduce the concept of $\mathbb F_{q^m}$-independence of $\mathbb F_q$-spaces and we show that $q$-matroids generalize this notion. As a consequence, the independent spaces of a representable $q$-matroid will be defined as the $\mathbb F_{q^m}$-independent subspaces of the $q$-system associated to an $\mathbb F_{q^m}$-linear rank-metric code. Moreover, we will further investigate the link between codes and matroids.
Denote by $\mathcal M_{g,n}$ the moduli space parametrizing a genus $g$ smooth complex curves with $n$-marked points. These moduli spaces are one of the central objects in modern science which appears in algebraic geometry, arithmetics, quantum physics… The challenging problem which we will discuss is the computation of the cohomology of $\mathcal M_{g,n}$ In my talk we will attack this problem from point of view of a triad: $$ (\mathcal M_{2g+n-1}^{\mathbb R},\mathcal M_{g,n},M_{g,n}^{trop}) $$ Here $\mathcal M_{2g+n-1}^{\mathbb R}$ is a moduli space of real curves and $M_{g,n}^{trop}$ is a moduli space of tropical curves. I will explain the correspondences between different moduli spaces of the triad. In particular we relate the cohomology of $\mathcal M_{g,n}$ to the cohomology of combinatorial objects called Kontsevich-Penner ribbon graph complex and the hairy Kontsevich graph complex. Further, we will discuss the correspondence between different structures on the elements of the triad. The talk is an overview of the works of K. Costello '07, S. Merkulov and T. Willwacher '15, M. Chan S. Galatius and S. Payne '18-'19, A. Andersson T. Willwacher and M. Zivkovič '20, T. Willwacher and S. Payne '21, A.K. '20-'22.
Differential polynomials of degree at most 1 annihilates univariate holonomic functions. In this talk, I will consider functions annihilated by differential polynomials of degree at most 2. It turns out that we can describe them with a finite amount of data. Therefore they provide a computational framework for a particular class of non-holonomic functions, including the generating functions of several sequences like the Bernoulli numbers, the Euler numbers, the numbers of alternating permutations of n letters, the evaluations of the zeta function at even integers, and many more.
A translation surface is a collection of polygons in the plane with parallel sides identified by translation to form a Riemann surface with a singular Euclidean structure. A saddle connection is a special type of closed geodesic. I will discuss recent work showing that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has quadratic asymptotic growth. No previous knowledge of translation surfaces or counting problems will be assumed. This is joint work with Jayadev Athreya and Howard Masur.
The subrank of a tensor is a value encoding to what extent a tensor is "stronger" than any tensor of a given rank. For this reason, tensors having large subrank play the role of universal objects for tensor rank, and find applications in numerous areas such as quantum physics and computational complexity. In this seminar, I will show that mild genericity properties on a tensor give strong lower bounds on its (asymptotic) subrank. I will emphasize connections with the notion of homomorphism duality, originated in graph theory, as well as the role of classical algebraic geometry and invariant theory. This is based on joint work with Matthias Christandl and Jeroen Zuiddam.
The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.
Consider the problem of $3D$ reconstruction in Computer Vision. Given $m$ images of a $3D$ scene, the reconstruction process starts with identifying the corresponding sets of points and lines, on the $m$ images. In this talk, I will give an algebraic understanding of line correspondences, I will describe the smallest algebraic set containing the image lines coming from the same line in $3D$, namely the line multiview variety, as well as illustrate it with an example, using Macaulay2. This is joint work with Paul Breiding, Felix Rydell, and Angélica Torres.
In the setting of normal form games, the Nash equilibrium analyses when no player can increase their expected payoffs by changing their strategy while assuming the other players have fixed strategies. In this case, each player acts independently, without any communication to the other players. In contrast, the concept of dependency equilibrium, introduced by philosopher Wolfgang Spohn in 2003, studies the case where the players simultaneously maximize their conditional expected payoffs. The Spohn variety is the algebraic interpretation of dependency equilibria. We view these two concepts of equilibria in terms of Bayesian networks. The Nash conditional independence curve (CI) is defined as the intersection of the Spohn variety with the statistical model of one-edge Bayesian networks. In other words, the Nash (CI) curve arises when only the choices of two players are dependent on each other. In this talk, we will explore certain algebro-geometric features of this curve as the genus, the degree, smoothness, etc. This is a joint-work with Irem Portakal.
This talk concerns statistical models in the n-simplex where the Maximum Likelihood Estimator (MLE) is a rational function. Eliana Duarte, Orlando Marigliano and Bernd Sturmfels recently proved that such models all arise as the image of a so-called Horn map. In their paper, they ask whether models with a rational MLE can be classified. We study the case where the models have dimension 1. In this case, after some simple reduction steps, such models correspond to outcomes of directed chipfiring games (or actually, more conveniently chipsplitting games) on a certain graph. We conjecture an upperbound on the size of these outcomes, which gives an upperbound on the degree of the corresponding models, and prove this conjecture for n
We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect that the triangle types have on curvature, and what can be revealed about these types from the curvature. Our results are established both by proof and numerical experiments, and shed light on the intricate geometry of the tropical projective torus. This is joint work with Anthea Monod.
A rank-r tensor is identifiable if it can be decomposed in a unique way as the sum of r elementary tensors. Rank-2 and rank-3 tensors are almost all identifiable with only few exceptions. In this talk I will present a complete classification on the identifiability of such small rank tensors.
In statistics, discrete probability distributions and gaussians centered at 0 are fundamental. The collection of discrete and centered gaussian distributions on n-variables can be modeled by the probability simplex and the positive definite cone of matrices respectively. A problem in statistics is to maximize the log-likelihood function restricted to a semi-algebraic subset of these models, given some statistical data. Transitioning to the complex numbers we may instead count the number of critical points, which we define to be the Maximum Likelihood Degree (ML-degree) of the corresponding subvariety. This concept is similar to the Euclidean Distance Degree (EDD) and yields an algebraic optimization problem. In the discrete case there are many nice results, such as the ML-degree being an Euler characteristic or the classification of all ML-degree 1 models. In my talk I will discuss ideas for proving analogues of these results in the Gaussian case.
Algebraic geometry has made great advances in the last two centuries. A particular role was played by enumerative geometry, where correct setting of moduli spaces found applications beyond mathematics. In my talk I would like to present a new work on applications of enumerative geometry providing a unified approach to fundamental invariants in algebraic statistics, combinatorics and topology. Achieving our results would not be possible without the fundamental work of De Concini, Huh, Laksov, Lascoux, Pragacz, Procesi and Sturmfels. The talk is based on joint works with Conner, Dinu, Manivel, Monin, Seynnaeve, Vodicka and Wisniewski.
While the concept of multiplicity is essential in the intersection theory, there is no such analogue for solutions of differential algebraic equations. In this talk I will motivate the definition of the multiplicity of a solution as the growth rate of the multiplicities of its truncations by considering the differential ideal of the fat point $x^m$. At the end I will briefly discuss some combinatoric connections between the multiplicity structure of the arc space of a fat point and Rogers-Ramanujan partition identities.
This is an ongoing project with Gleb Pogudin.
Intersection bodies are classical objects from convex geometry. But using tools from combinatorics and real algebraic geometry, what more can we say about these objects?
In this talk we show that the intersection body of a polytope is always semialgebraic. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for their degree. This is joint work with Katalin Berlow, Chiara Meroni and Isabelle Shankar.
In science and engineering we regularly face hard, nonlinear polynomial optimization problems. Solving these problems is essentially equivalent to certifying nonnegativity of real, multivariate polynomials – a fundamental problem in real algebraic geometry since the 19th century. It is well-known that in general, this problem is very hard; therefore, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. A standard nonnegativity certificate is given by sums of squares (SOS), which can be detected by semidefinite programming (SDP). This SOS/SDP approach, however, has some issues, especially in practice if the problem has many variables or high degree.
In this talk, I will introduce sums of nonnegative circuit (SONC) polynomials - a new class of nonnegativity certificates for real, multivariate polynomials. These certificates are independent of sums of squares. I will present some structural results of SONC polynomials and I will provide an overview of polynomial optimization via SONC polynomials.
In a joint work with Jose Alejandro Samper, we study the cone of completely positive (cp) matrices for the first interesting case n=5. This is a semialgebraic set, which means that the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.
The size of an explicit representation of a given rational point on an algebraic curve is captured by its canonical height. However, the canonical height is defined through the dynamics on the Jacobian and is not particularly accessible to computation. In 1984, Faltings related the canonical height to the transcendental "self-intersection" number of the point, which was recently used by van Bommel-- Holmes--Müller (2020) to give a general algorithm to compute heights. The corresponding notion for heights in higher dimensions is inaccessible to computation. We present a new method for computing heights that promises to generalize well to higher dimensions. This is joint work with Spencer Bloch and Robin de Jong.
The toric ideal associated to a finite graph is obtained by taking the kernel of the monomial map that is defined by the edges of the graph. Equivalently one obtains a toric variety by defining edge cones (or edge polytopes) where the extremal rays (or vertices) are the columns of the incidence matrix of the graph. In this talk, we explain the interplay between graphs and their associated toric varieties appearing in different areas such as Fano, (matrix) Schubert and Kazhdan-Lusztig varieties.
Polytopes with rational linear precision are of interest in the Geometric Modeling community because of their approximation properties and it is an open question to classify them in dimension d > 2. This classification question is closely related to discrete statistical models with rational maximum likelihood estimators. In this talk I will introduce a new family of lattice polytopes with rational linear precision in higher dimension using techniques from Algebraic Statistics. This is joint work with Isobel Davies (OVGU), Irem Portakal (MPI MIS) and Miruna-Stefana Sorea (SISSA).
For curves over the field of p-adic numbers, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally, and Vologodsky integrals with desirable number-theoretic properties. These integrals have the advantage of being insensitive to the reduction type at p, but are known to coincide with Coleman integrals in the case of good reduction. Moreover, there are practical algorithms available to compute Coleman integrals.
Berkovich-Coleman and Vologodsky integrals, however, differ in general. In this talk, we give a formula for passing between them. To do so, we use combinatorial ideas informed by tropical geometry. We also introduce algorithms for computing Berkovich-Coleman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by certain open spaces, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the Berkovich-Coleman integrals into Vologodsky integrals using our formula. We illustrate our algorithm with a numerical example.
This talk is partly based on joint work with Eric Katz.
Conditional independence is a ternary relation on subsets of a finite vector of random variables $X$. A \textbf{CI statement} $(ij|K)$ asserts that "whenever the outcome of all the variables $X_k$, $k$ in $K$, is known, learning the outcome of $X_i$ provides no further information on $X_j$". These relations are highly structured, in particular under assumptions about the joint distribution. The goal is to describe this by \textbf{CI inference rules}: given that certain CI statements hold, which other (disjunctions of) CI statements are implied under the distribution assumption?This talk is about regular Gaussian distributions. In this case, conditional independence has an algebraic characterization in terms of subdeterminants of the covariance matrix and inference, a discrete problem by nature, becomes a geometric question about the vanishing of very special polynomials on very special varieties inside the cone of positive-definite matrices.In the first part of the talk, I show that the space of counterexamples to a (wrong) inference formula can be "difficult" by multiple measures. In particular, proving inference formulas wrong is polynomial-time equivalent to the existential theory of the reals. In the second part, I report on practical approximations to the inference problem and computational results on the way of classifying all Gaussian CI structures on five random variables.
To model evolution, one usually assumes that DNA sequences evolve according to a Markov process on a phylogenetic tree ruled by a model of nucleotide substitutions. This allows to define a joint distribution at the leaves of the trees and to obtain polynomial relationships among these joint probabilities. The study of these polynomials and of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees.
However, not all points in these algebraic varieties have biological sense. In this talk, we will discuss the importance of studying the subset of these varieties that has biological sense and we will prove that, in some cases, considering this subset is fundamental for the phylogenetic reconstruction problem. Finally, we will present a new phylogenetic quartet reconstruction method which is based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet tree.
We are interested in counting cubic hypersurfaces in projective n-space tangent to enough many points and lines. Paolo Aluffi explored the case for plane cubic curves. Starting from his work we construct a 1-complete variety of cubic hypersurfaces by a sequence of five blow-ups over the space parametrizing the cubics. The problem is then reduced to compute five Segre classes by climbing the sequence of blow-ups. This is an ongoing project with Mara Belotti, Alessandro Danelon e Andreas Kretschmer.
We investigate a particular class of semialgebraic zonoids called discotopes. Our aim is to study the algebraic properties of the boundary of such convex bodies. We analyze their face structure and discuss some geometric features. This is a joint work with Fulvio Gesmundo.
We investigate linear PDEs with constant coefficients. These are viewed as constraints for nonlinear variational problems arising in Continuum Mechanics, most properties of which are well understood in the case of constant rank'' operators. We use algebraic properties of the linear PDE to derive a first lower semicontinuity result for variational integrals under non-constant rank constraints.
We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis--Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.
Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. In this talk, we study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We present determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We show how to use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with a new eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP. This talk is based on joint work with Jean-Charles Faugère, Angelos Mantzaflaris, and Elias Tsigaridas.
This talk is about the strength of homogeneous polynomials, which has only been defined a few years ago.What is the strength of a polynomial?Why look at it?How do you compute it?Is bounded strength a closed condition?What is the strength of a generic polynomial?I will answer some of these questions.
An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. In the talk, we show that there are polynomial time algorithms for all three of the aforementioned problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). The talk is based on a joint work with P. Bürgisser, V. Makam, M. Walter and A. Wigderson.
In recent years, Computational Invariant Theory has seen significant progress in optimization techniques: so-called scaling algorithms approximately minimize the norm along an orbit under a group action. Such algorithms also give rise to methods for /deciding /null-cone membership, i.e. whether zero is in a given orbit closure. Both computational problems have versatile applications in mathematics, physics, statistics and computer science. However, only for certain group actions polynomial time algorithms are known.
In this talk, we give a short introduction to these computational problems and provide exponential bounds on certain complexity parameters. These results explain why current techniques are, in general, only known to run in exponential time and strongly motivate the search for new scaling algorithms.
This is joint work with Cole Franks, see arXiv:2102.06652, https://arxiv.org/abs/2102.06652.
The study of non-negative polynomials is motivated by the obvious fact that the value at a global minimum of a real polynomial $f$ is the maximal value $c$ such that $f-c$ is globally non-negative. This shows its connection to optimization. Similarly, a local minimum $x_0$ of $f$ induces the polynomial $f-f(x_0)$ which takes value $0$ and is locally non-negative at $x_0$.I will present results from my PhD thesis on the convex cone of locally non-negative polynomials. We will see geometric interpretations and examples of faces of this cone, some general theory of cones in infinite-dimensional vector spaces and classifications of faces using tools from singularity theory. I will also give a short outlook on an application to sums of squares in real formal power series rings.
For each hyperbolic polynomial h, there is an associated closed convex cone called the hyperbolicity cone of h, whose interior contains all the directions e for which h is hyperbolic. Moreover, a convex cone is called spectrahedral, if it can be described by linear matrix inequalities with symmetric matrices. Is every hyperbolicity cone spectrahedral? This is the question generalized Lax conjecture considers and posits.
Choe et. al. in 2004 showed that the support of each homogeneous multiaffine polynomial with the half-plane property (such a polynomial is hyperbolic) is the collection of bases of some matroid M. Their result lets us switch to the combinatorial world, search for matroids corresponding to a hyperbolic polynomial, and consider the spectrahedral representability in that setting.
In this talk, we take this matroid theoretic approach, and present our results on the spectrahedral representability being closed under taking minors. We continue with the classification of matroids on 8 elements with respect to the half-plane property.
We study non-Gaussian graphical models from a perspective of algebraic statistics. Our focus is on algebraic relations among second and third moments in graphical models given by linear structural equations.We show that when the graph is a tree these relations form a toric ideal. From the covariance matrix and the third moment tensor, we construct explicit matrices (associated to treks and multi-treks) whose $2\times 2$ minors generate the vanishing ideal of the model.This is a joint work with Carlos Améndola, Mathias Drton, Alexandros Grosdos and Elina Robeva.
We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We study the Hirota variety associated to familiar Delaunay polytopes, in particular characterizing it for the g-cube.
If time permits, we will also compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and present an algorithm that finds a soliton solution from a rational nodal curve.
Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of 3-polytopes with edges tangent to the unit sphere. We introduce two notions of algebraic degree for such constrained realizations and we compute them for some classes of polytopes. This is joint work with Michael Joswig and Marta Panizzut.
Pairs [toric idealmonomial algebra] associated to monomial maps between polynomial rings in infinitely many variables will be considered. We will introduce the concept of finiteness up to a shift operator on the indices of variables. In the Noetherian setting these two objects behave very similarly. Surprisingly, they do not necessary behave the same in the infinite world. For instance, it will be shown that there are monomial algebras that are finitely generated up to our shift operator, and their respective ideals require generators of any degree. The second part of the talk will be on computing the rational form of an Equivariant Hilbert series for these objects using regular languages and finite automata. If time permits, inspired by Segre and tensor products we will end on definitions of Segre tensor languages.
Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will focus on a concrete case used to state the theorem and highlight the proof strategy. We will also discuss some ongoing work and ideas which advertise the generality and strength of this argument.
Mathematical objects of all types and flavors admit a notion of a 'model'. These are objects of the same type as our starting object of interest, which are better behaved, and are useful to obtain qualitative and quantitative information about it. Minimal models are "optimally small'' models that usually lend themselves to computation. Determining the minimal model of an object, when it exists, is generically a difficult task.
In this talk, I will explain how to obtain the minimal model of an associative algebra defined by monomial relations, as in 1804.01435. I will survey related results, and will mention some open questions and conjectures that emerged from 1804.01435 and related work 1909.00487 with Dotsenko and Gelinas.
I will give a gentle introduction (with computations!) to some ideas related to the KP equation. We will first see pseudo-differential operators, which are linear operators (on a suitable space of functions) containing both differential and integral parts. Next, we will study the ring of commuting differential operators with a given differential operator P, and finally we will see some basic algebraic data associated with P. While this is only a glimpse into a very rich theory, it sets the stage for very interesting objects such as the Sato Grassmannian, Lax pairs, Jacobians, algebraic curves, etc.
Gaussian double Markovian models consist of covariance matrices with specified zeros and specified zeros in their inverses. Geometrically, these are intersections of graphical models with inverse graphical models. We describe the semi-algebraic geometry of these models, in particular their dimension, smoothness, connectedness, and vanishing ideals. We give a geometric proof of an exponential family heuristic for a smoothness criterion of Zwiernik. We also continue investigations of singular loci initiated by Drton and Xiao.
This is joint work with Tobias Boege, Andreas Kretschmer, and Frank Röttger.
In calculus, one learns how to deal with Taylor series of smooth functions. How to apply that theory to regular functions on an algebraic variety? In this talk, I recall the algebraic counterpart of Taylor series: arc and jet schemes of algebraic varieties. I will explain how those are related to mapping a line into a variety.
Coding theory can be seen as the theory of subsets/subspaces of a certain metric space. The most studied and known setting is undoubtedly the Hamming metric. There, one consider a finite dimensional vector space over a (finite) field K, and the distance between two vectors is the number of entries in which they differ. In the last decades, the attention of many researchers has been shifted to rank-metric codes, i.e. linear spaces of matrices over a field K in which the metric considered is given by the rank. Very recently, the sum-rank metric has attracted many people: here the ambient space consists of t-uples of matrices of fixed sizes over a field K, while the distance between two tuples is obtained by adding up the ranks of the differences of the constituent matrices. However, all these metric spaces can be considered as special cases of a more general setting: spaces of matrices with restricted entries, in which the distance of two matrices is the rank of their difference. In this talk I will present a unifying framework for all these metric spaces, which connects them with special spaces of matrices. In particular, I will show how these metric spaces are isometric to suitable quotients of skew polynomial rings, and how to construct linear spaces of matrices with restricted entries in which every nonzero matrix has high rank.
It was observed by Pukhlikov and Khovanskii that the BKK theorem implies that the volume polynomial on the space of polytopes is the Macaulay generator of the cohomology ring of a smooth projective toric variety. This provides a way to express the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. The crucial ingredient of this observation is an explicit expression for the Macaulay generator of graded Gorenstein algebras generated in degree 1.
In my talk I will explain this construction in detail, then I will tell about recent results on explicit expression for the Macauley generator of an arbitrary algebra with Gorenstein duality. Finally, if time permits, I will show how these results yield to the computation of the cohomology rings of more general classes of algebraic varieties.
Sum of squares (SOS) relaxations are often used to certify nonnegativity of polynomials and are equivalent to solving a semidefinite program (SDP). The feasible region of the SDP for a given polynomial is the Gram Spectrahedron. For symmetric polynomials, there are reductions to the problem size that can be done using tools from representation theory. In joint work with Serkan Hosten and Alexander Heaton, we investigate the geometric structure of the spectrahedra that arise in the study of symmetric SOS polynomials, the Symmetry Adapted PSD cone and the Symmetry Adapted Gram Spectrahedron.
In this talk I will speak about ongoing work with Hana Melánová and Bernd Sturmfels, in which we study the problem of recovering a collection of n numbers from the evaluation of m power sums. The polynomial system corresponds to intersecting Fermat hypersurfaces, and it can be underconstrained (m < n), square (m=n), or overconstrained (m > n). Questions that we ask are for example, when is recovery possible? If it is possible, is it unique? If it is not unique, can we give an upper bound for the number of solutions? I will present some results, and many more conjectures.
In statistics, linear concentration model is given by the linear space of the symmetric matrices. Given the linear space L, there are two interesting numbers: The degree of variety $L^{-1}$, obtained by inverting all matrices in L and ML-degree of the model. It was shown that for general space L these two numbers are the same. However, this is not true for some specific spaces L. In this talk we will discuss such a case when L is the space for the Gaussian graphical model for the cycle. In this case these two numbers are different. We will focus on the degree of the variety $L^{-1}$ for which there is an explicit formula, conjectured by Sturmfels and Uhler, which we were able to prove. The proof is based on the intersection theory in the space of complete quadrics, intersection theory in Grassmannian and a lot of combinatorics.
We consider the tropicalization of determinantal varieties of matrices of rank at most 2 and determine which cones correspond to the tropicalization of positive matrices. We give a combinatorial description of this positive part in terms of tropically collinear point configurations, bicolored phylogenetic trees and perfect matchings in bipartite graphs.
This is joint work in progress with Georg Loho and Rainer Sinn.
Multiparameter eigenvalue problems can be seen both as generalizations of eigenvalue problems and of linear systems of equations. We will discuss how real symmetric eigenvalue problems generalize to the multiparameter case and how we can use their properties to solve them numerically. This is joint work with Yuji Nakatsukasa.
Bruhat Tits buildings are central objects in the theory of algebraic groups that carry a rich combinatorial structure. A central role in the combinatorial investigation of Bruhat-Tits buildings is played by various different notions of convexity. In this talk, we will introduce these different notions, relate them to tropical geometry and the theory of orders.
It is a classical fact that the sequence of powers of a symmetric matrix converges, up to scaling, to the orthogonal projection onto the eigenspace determined by its largest eigenvalue. We explore analogous statements for symmetric tensors, where the matrix powering operation is replaced by tensor contractions encoded by the combinatorics of a graph. I will discuss some positive and negative recent results in this direction. This is joint work with A. Uschmajew.
In an undergraduate differential equations course we learn to solve a homogeneous linear ordinary differential equation with constant coefficients by finding roots of the characteristic polynomial. Thus the problem of solving an ODE is reduced to factoring a univariate polynomial. A generalization of this was found in the 1960s for systems of linear PDE. The Fundamental Theorem by Ehrenpreis and Palamodov asserts that solutions to a PDE system can be represented by a finite sum of integrals over some algebraic variety. This representation can be used describe a primary decomposition of an ideal or module. This talk presents the main historical results, along with recent algorithms.
We study families of faces for convex semi-algebraic sets via the normal cycle. This is a semi-algebraic set similar to the conormal variety in projective duality theory. Families of faces are represented by a finite list of patches. This is joint work with Daniel Plaumann and Jannik Wesner.
The Pauli exclusion principle consists of linear constraints on the occupation numbers of electrons (namely being between 0 and 1). We now consider the spectra of reduced density matrices and compute explicit necessary linear inequalities satisfied by said spectra. Each linear inequality can be thought of as a generalized exclusion principle. We study the polytope defined by these inequalities, which turn out to have nice combinatorial properties. This is based on joint work with JP. Labbe, J.Liebert, A.Padrol, E.Philippe and C.Schilling.
We introduce a new algorithm for enumerating chambers of a hyperplane arrangement which exploits its underlying symmetry group. Our algorithm counts chambers as a byproduct of computing Betti numbers. We showcase our implementation in OSCAR on examples coming from hyperplane arrangements with applications to physics and computer science.
A staged tree model is a statistical model which can be represented by a tree. To each staged tree we associate a prime ideal of homogeneous polynomials, which provides an algebraic description of the same model. In this talk we will discuss algebraic properties of this ideal, and how they can be read off from the tree. In particular we want to understand if the ideal is toric, possibly after a linear change of coordinates. This is joint work with Christiane Görgen and Aida Maraj.
The theta function of a smooth algebraic curve provides solutions to the KP equation in mathematical physics. The theta function is highly transcendental function, but this can change when the curve becomes singular. I will present a classification of those singular curves whose theta function is polynomial, and prove that give rational solutions to the KP equation. In particular, I'll try to explain how everything essentially follows from Abel's theorem. This is joint work with John B. Little and Türkü Çelik.
The number of inversions or descents of a random permutation in a large symmetric group is asymptotically normally distributed. We discuss extensions of this principle to arbitrary families of finite Coxeter groups of increasing rank. As a prerequisite we find uniform formulas for the means and variances in terms of Coxeter group data. The main gadget for central limit theorems is the Lindeberg—Feller theorem for triangular arrays. Transferring the Lindeberg condition to the combinatorial setting, one finds that the validity of a central limit theorem depends on the growth of the dihedral subgroups in the sequence.
A smooth plane quartic defined over the complex numbers has precisely 28 bitangents. This result goes back to Pluecker. In the tropical world, the situation is different. One can define equivalence classes of tropical bitangents of which there are seven, and each has 4 lifts over the complex numbers. Over the reals, we can have 4, 8, 16 or 28 bitangents. The avoidance locus of a real quartic is the set in the dual plane consisting of all lines which do not meet the quartic. Every connected component of the avoidance locus has precisely 4 bitangents in its closure. For any field k of characteristic not equal to 2 and with a non-Archimedean valuation which allows us to tropicalize, we show that a tropical bitangent class of a quartic either has 0 or 4 lifts over k. This way of grouping into sets of 4 which exists tropically and over the reals is intimately connected: roughly, tropical bitangent classes can be viewed as tropicalizations of closures of connected components of the avoidance locus. Arithmetic counts offer a bridge connecting real and complex counts, and we investigate how tropical geometry can be used to study this bridge.
This talk is based on joint work with Maria Angelica Cueto, and on joint work in progress with Sam Payne and Kristin Shaw.
In this talk, I will introduce the audience to diagram algebras and their numerous connections to other areas of mathematics and mathematical physics. We will discuss the appearance of diagram algebras in statistical physics (loop models, Potts models), quantum group theory, combinatorics and stochastic processes on graphs. In particular, in the second half of the presentation, I will explain recent results on diagram algebras and their connection to random lattice paths and random walks on trees.
A random planar map is a canonical model for a discrete random surface which is studied in probability, combinatorics, mathematical physics, and geometry. Liouville quantum gravity is a canonical model for a random 2d Riemannian manifold with roots in the physics literature. In a joint work with Xin Sun we prove a strong relationship between these two natural models for random surfaces. Namely, we prove that the random planar map converges in the scaling limit to Liouville quantum gravity under a discrete conformal embedding which we call the Cardy embedding.
Topological Data Analysis analyzes the shape of data by topological methods. The main tool is persistent homology. In the one-parameter setting, classical theorems from Algebra allow to associate so-called barcodes, from which one easily reads topological features of the dataset. The multiparameter case is in need of advancing the algebraic tools behind the scenes. In an ongoing project together with Valeria Bertini and Christian Lehn, we are working on a classification of quotients of free multigraded modules using Quot schemes.
In this talk, I give a friendly introduction to Topological Data Analysis and outline how the aforementioned moduli spaces enter the stage.
Interacting particle systems such as Asymmetric Simple Exclusion Process (=ASEP) form an interesting and well-studied class of stochastic systems.
It turns out that multi-species versions of several of interacting particle systems, including ASEP, can be interpreted as random walks on Hecke algebras. In the talk I will discuss this connection and its recent probabilistic applications. No preliminary knowledge about the topic is required.
We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. Even more interesting than that bound are various examples to illustrate differences between Gröbner and tropical bases.
Joint work with Benjamin Schröter.
We discover a geometric property of the space of tensors of fixed multilinear (Tucker) rank. Namely, we show that real tensors of fixed multilinear rank form a minimal submanifold of the Euclidean space of all tensors of given format endowed with the Frobenius inner product. This is joint work with Khazhgali Kozhasov and Lorenzo Venturello.
In this talk, I will explain how one can use persistent homology, a method from topological data analysis, to study astrophysics data statistically.
In joint work with Sven Heydenreich and Joachim Harnois-Déraps, we study data coming from weak gravitational lensing, i.e. the deflection of light from background galaxies by the large-scale structure of the Universe. We show that working with persistent homology yields better results than currently used standard methods in this field. After giving some background on the physical motivation, I will explain the pipeline of our analysis and in particular focus on the topological methods.
The goal of this talk is to discuss hyperbolic hypersurfaces, an extremal real topological type. A particular emphasis is on singular examples and I want to discuss one source of them, namely a method to construct them by considering secant varieties of real curves with the maximal number of connected components (aka M-curves).
This is based on joint work with Mario Kummer.
Steinitz's problem asks whether a triangulated sphere is realizable geometrically as the boundary of a convex polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their creation more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with little success. In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of an approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.
The commuting variety is a well-studied object in algebraic geometry whose points are pairs of matrices that commute with one another. In this talk I present a generalization of the commuting variety by using the notion of commuting distance of matrices, which counts how many nonscalar matrices are required to form a commuting chain between two given matrices. I will prove that over any algebraically closed or real closed field, the set of pairs of matrices with bounded commuting distance forms an affine variety. I will also discuss many open problems about these varieties, and present preliminary results in these directions. This is based on joint work with Madeleine Elyze and Alexander Guterman.
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.
I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.
Gröbner bases are one the most powerful tools in algorithmic nonlinear algebra and a standard tool to solve polynomial systems. Their computation is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example, several problems in computer-aided design, robotics, vision, biology, kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms.
In this talk, we discuss how to solve polynomial systems faster by exploiting their sparsity. We do so by introducing tools coming from toric geometry to the Gröbner-basis framework. Our strategy is to perform our computations over multigraded semigroup algebras associated to the Newton polytopes of the input systems. We present the first algorithm to compute Gröbner bases for sparse systems that, under regularity assumptions, performs no redundant computations. Additionally, we discuss the complexity of our approach, its dependence on the combinatorics of the input polytopes and how, for particular families of sparse systems, we can use the multigraded Castelnuovo-Mumford regularity to improve our complexity bounds.
I will describe recent research on the intersection of algebraic geometry and computer vision, beginning with a discussion of the basics of 3D computer vision from a geometric point of view. The talk will touch on joint work with Kathlén Kohn, Viktor Korotynskiy, Anton Leykin, Tomas Pajdla, and Maggie Regan. The focus is on solving "minimal cases" for the problem of reconstruction of a configuration of points and lines in space. Under various natural hypotheses we can enumerate all minimal problems (basic dimension counting) and compute the number of complex solutions (symbolic/numerical methods.) Time-permitting, I will explain a fun connection between the five-point problem (a workhorse of modern RANSAC-based reconstruction pipelines) and the Coxeter group D_10, and how this relates to the general program of whether or not minimal problems "simplify".
We study the usual torus action on Kazhdan-Lusztig varieties and examine the complexity of the this torus action by utilizing certain simple directed graphs. This is a joint work with Maria Donten-Bury and Laura Escobar.
Solutions to sparse polynomial systems can be viewed as fibres of a branched cover determined by the support of the equations. A system is decomposable if this branched cover factors as a composition of nontrivial branched covers on an open set. This leads to a method of solving sparse polynomial systems by completely factoring the branched cover and iteratively solving simpler polynomial systems as fibres of these factors.
Homotopies are useful numerical methods for solving systems of polynomial equations. Embedded toric degenerations are one source for homotopy algorithms. In particular, if a projective variety has a toric degeneration, then linear sections of that variety can be optimally computed using the polyhedral homotopy. Any variety whose coordinate ring has a finite Khovanskii basis is known to have a toric degeneration. We provide embeddings for this Khovanskii toric degeneration to compute general linear sections of the variety. This is joint work with Michael Burr and Frank Sottile.
One of the most important invariants one can associate to a matrix is its rank, which expresses how many pairs of vectors you need to write down a formula for the matrix. Consider infinite-by-infinite matrices. For such a matrix, we define its rank to be the supremum of all its finite-by-finite submatrices. This can be finite, which is equivalent to the matrix A being able to be expressed using finitely many infinite vectors. Or else it is infinite, which turns out to be equivalent to stating that the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all infinite-by-infinite matrices.
For polynomial series, their strength fulfil a similar role. We define the strength of a polynomial series to be the infimum number of pairs of lower degree series needed to write down a formula for it. It is then true that the strength of a polynomial series is infinite if and only if the set of series obtained from it by finitely many substitutions is Zarisky-dense in the space it lives in. Both infinite-by-infinite matrices and polynomial series are examples of the following dichotomy: either you can express them using a finite amount of lower-dimensional data or their orbit under some group is dense.
This talk is about joint work with Jan Draisma, Rob Eggermont and Andrew Snowden that generalizes this statement to all finite-degree polynomial functors.
The notion of discrete Ricci curvature for graphs was introduced in 1999 by Schmuckenschlager, it is part of the attempt to translate some notions of Riemannian geometry to graph theory. Among some more recent approaches to the discrete Ricci curvature there is a paper from Klartag et al. where the curvature of various Cayley graphs is computed. From these results arose the interest for the discrete Ricci curvature of graphs in Coxeter theory. During the talk I will recall the definition of discrete Ricci curvature together with some basic properties. I will then compute it for graphs associated to Coxeter groups, namely Bruhat graphs and weak order graphs.
The purpose of this talk is to understand a recent result obtained by Etienne Ghys, namely: which are the permutations obtained by intersecting the graphs of a family of polynomials in one real variable that pass through a common zero at the origin. We will discover surprising connections between different geometric and combinatorial objects.
Tropical convex hulls of sets of points in R^n are called tropical polytopes. These have been deeply study in the past years in various areas of research. In this talk I will illustrate results concerning instead tropical convex hulls of convex sets. I will illustrate their properties and apply the obtained result to prove the tropical analogue of a classical inequality for the degree of a curve.
The Maximum Likelihood Degree (ML-degree) measures algebraic complexity of a statistical model. This invariant is closely related to the multi-degree of the graph of a gradient map and has been an object of study (under different names) in singularity theory, algebraic geometry, commutative algebra and statistics. Apart from a few cases the formulas for the invariant are not known even in the case of general Gaussian models. Based on a joint work with Monin and Wisniewski, we provide two new approaches to this old problem. Both are related to geometry of the Lagrangian Grassmannian and tori actions. One relates to Schubert calculus and one to Gromov-Witten like invariants.
Quantitative models and quantitative analysis in Computer Science are receiving increased attention. The goal of this talk is to investigate quantitative automata and quantitative logics. Weighted automata on finite words have already been investigated in seminal work of Schützenberger (1961). They consist of classical finite automata in which the transitions carry weights. These weights may model, e.g., the cost, the consumption of resources, or the reliability or probability of the successful execution of the transitions. This concept soon developed a flourishing theory. We investigate weighted automata and their relationship to weighted logics. For this, we present syntax and semantics of a quantitative logic; the semantics counts ‘how often’ a formula is true in a given word. Our main result, extending the classical result of Büchi, shows that if the weights are taken from an arbitrary semiring, then weighted automata and a syntactically defined fragment of our weighted logic are expressively equivalent. A corresponding result holds for infinite words. Moreover, this extends to quantitative automata investigated by Henzinger et al. with (non-semiring) average type behaviors, or with discounting or limit average objectives for infinite words.
We introduce unimodular triangulations, covers and the integer decomposition property of lattice polytopes, and briefly touch on motivations for their study. We then show that the following special classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds and the class of Cayley sums of two polytopes where one is a weak Minkowski summand of the other. Based on joint work with Francisco Santos.
Multiparameter persistence is a generalisation of persistent homology, one of the main tools in topological data analysis. In this talk I will introduce multiparameter persistence modules and give an overview of invariants used to study them.
Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph, naturally generalizing the ideals of 2-minors of a generic matrix with two rows. They also arise in Algebraic Statistics in the context of conditional independence ideals. We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne’s Connectedness theorem. As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro.
This is a joint work with Davide Bolognini and Francesco Strazzanti.
A reaction network is a finite oriented graph embedded in Euclidean space. Dynamical systems generated by reaction networks can have arbitrary polynomial (or rational) right-hand-side. This creates a strong relationship between the study of reaction networks and the study of the set of solutions of some associated polynomial equations. In particular, many important questions about dynamical properties of reaction network models can be translated into questions about polynomial equations generated by the corresponding oriented graph.
For example, if some special polynomial equations have positive solutions, then the associated reaction network has a globally attracting fixed point. Similarly, if a special polynomial is positive everywhere, then the associated reaction network has at most one fixed point.
We will discuss some of these connections and several open problems.
Moments and cumulants are quantities that measure the shape of statistical distributions and have recently gained interest from an algebraic and combinatorial point of view. In this talk we focus on two problems that are associated with them. First we try to find the moment ideal, that is the ideal parametrized by the moments of a distribution. Further, we explain how mixtures of distributions give rise to secants of the varieties in question, while local mixing corresponds to tangents. The second problem is about parameter recovery from moments or cumulants. We explain why cumulants are better suited for this job when using computer algebra. We provide concrete examples with distributions coming from exponential families and Diracs.
I would like to discuss toric geometry methods which can be used for a variety admitting torus action whose rank is smaller than the dimension of the variety in question. These techniques has been used in a project regarding LeBrun-Salamon conjecture, joint with Jaroslaw Buczynski, Andrzej Weber, and Eleonora Romano.
My talk will quickly survey current trends of the study on (Castelnuovo--Mumford) regularity of monomial ideals. Especially, regularity of edge ideals arising from finite simple graphs and of their powers will be discussed. No special knowledge will be required to understand my talk.
We describe a generalization of the Sums-of-AM/GM Exponential (SAGE) relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that relative entropy based SAGE certificates conveniently and transparently blend with convex duality, in a manner that Sums-of-Squares certificates do not. This more general approach not only retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), but also inspires a novel perspective-based method of solution recovery.
The idea of studying rational maps by looking at the syzygies of the base ideal is a relatively new idea that has now become an important research topic. In this talk, we will discuss some recent results that lead to birationality criteria and formulas for the degree of rational maps that depend on the algebraic properties of the syzygies of the base ideal. In a related way, we will also discuss some results in the problem of specializing the coefficients of a rational map.
Oftentimes researchers and practitioners may be interested in learning not only a representative of a Markov equivalence class (MEC) of graphs, but a specific member of the class. In the case of DAGs, the classic approach to this problem is to sample data from an interventional setting and use this data to refine MECs of DAGs into smaller sets. This process naturally relies on combinatorial interpretations of invariance properties of distributions Markov to DAGs that can be used to define these refined interventional MECs of DAGs. Recently, Yang et al. (2018) gave a graphical characterization of this interventional Markov equivalence for DAG models that relates to the global Markov properties of DAGs. Using this idea, we will extend the notion of interventional Markov equivalence using global Markov properties of loopless mixed graphs and generalize their graphical characterization to ancestral graphs. On the other hand, we will also extend the notion of interventional Markov equivalence via invariance properties of distributions Markov to acyclic directed mixed graphs. Finally, we will see that these two generalizations coincide at their intersection; i.e., for directed ancestral graphs, thereby completely generalizing the theory of Yang et al. (2018) to this setting.
The structure and behavior of real algebraic varieties can be insightful for real world applications where computations over the complex numbers are not meaningful. First, I will present on a new definition of monodromy action over R which encodes tiered characteristics regarding real solutions. Examples will be given to show the benefits of this definition over a naive extension of the monodromy group (over C). In addition, an application in kinematics will be discussed to highlight the computational method and impact on calibration. Next, I will discuss a method for computing topological information such as Euler characteristic, genus, Betti numbers, and the generators of the fundamental group for smooth, compact, and orientable real algebraic surfaces. This underlying approach is via a cell decomposition computed using numerical algebraic geometry from the software Bertini_Real and then creating a simplicial complex modelling the real algebraic surface. The software Javaplex is then used to compute the desired topological information. Several examples will be discussed to demonstrate the approach.
This talk will be about interactions between C*-algebras, Topological Dynamics, and Group Theory, some of which are classical, while others emerged recently. I will give a brief (and friendly) overview, and discuss some future directions.
The signature of a path is a sequence of iterated coordinate integrals along the path. We aim at reconstructing a path from its signature. In the special case of lattice paths, one can obtain exact recovery based on a simple algebraic observation. For general continuously differentiable curves, we develop an explicit procedure that allows to reconstruct the path via piecewise linear approximations. The errors in the approximation can be quantified in terms of the level of signature used and modulus of continuity of the derivative of the path. The main idea is inspired by the simple procedure for lattice paths, but with serious involvement of analysis. A key ingredient is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales.
We propose a new algorithm for spectral learning of Hidden Markov Models (HMM). In contrast to the standard approach, we do not estimate the parameters of the HMM directly, but construct an estimate for the joint probability distribution. The idea is based on the representation of a joint probability distribution as an N-th-order tensor with low ranks represented in the tensor train (TT) format. Using TT-format, we get an approximation by minimizing the Frobenius distance between the empirical joint probability distribution and tensors with low TT-ranks with core tensors normalization constraints. We propose an algorithm for the solution of the optimization problem that is based on the alternating least squares (ALS) approach and develop its fast version for sparse tensors.
Tree model is one of the central objects in phylogenetics. A group-based model is a tree model where the input parameters are $G$-invariant. We will discuss mainly Kimura 3-parameter model which is a group model with underlying group $\Mathbb Z_2\times \Mathbb Z_2$. The varieties associated to this model are toric and there is an explicit description by family of polytopes associated to these varieties. Thus one can study properties of these varieties by studying properties of family of polytopes. We show normality of these polytopes meaning that the associated projective toric varieties are projectively normal.
A random geometric complex on a compact Riemannian manifold of dimension m is a union of balls centered at random points on the manifold — randomness comes from sampling the points from the uniform distribution. The radius r of these balls are chosen to scale together with the number n of sampled points: when $r=O(n^{-1/m})$ the geometric complex is in the thermodynamic regime. This is the regime when the topology of the complex is the richest. In this talk I will prove that when n goes to infinity the topological types of the components of this complex, appropriately rescaled, have a limit distribution. (Similar statements have recently been proved by Sarnak and Wigman for the topologies of nodal domains of random harmonics.) Of particular interest is the one-skeleton of this complex, which is a random geometric graph: in this case the existence of the previous limit implies the existence of a limit for the spectrum of this graph, which has interesting connections with the spectrum of the manifold.(This is based on joint work with A. Auffinger, E. Lundberg)
For a finite group W acting linearly on a polynomial ring R, and any subgroup K < W, we define the invariant coinvariant ring $R^K_W$ to be the quotient of the ring of K-invariant polynomials by the ideal generated by the W-invariant polynomials. In case W is a Weyl group and K is a parabolic subgroup the invariant coinvariant ring can be identified with the cohomology ring of a smooth complex projective manifold called a Grassmannian. These cohomology rings have nice algebraic properties, e.g. Poincaré duality, strong Lefschetz, Schubert calculus, and it seems natural to ask which other group pairs K < W have invariant coinvariant rings with these properties. It turns out that if K and W are both complex reflection groups, then $R^K_W$ always satisfies Poincaré duality, whereas strong Lefschetz can fail, even if K is "parabolic" (we conjecture this does not happen in the real case). Moreover in the complex case, a working Schubert calculus seems to be lacking, even in the simplest cases. I will attempt to fill in the details of this story, with plenty of examples, and then describe a connection to the combinatorics of partitions, and a remarkable conjecture of G. Almkvist.
We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.
The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this nonconvexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.
I will consider the problem of counting the number of points on the m-dimensional Sphere at which a homogenous polynomial attains a given nondegenerate singularity, i.e. points where the r-jet prolongation of the polynomial meets transversally some given semialgebraic submanifold of the space of r-jets. I will present a recent result, obtained jointly with Antonio Lerario, which establishes that in the Kostlan case, under reasonable hypothesis on the singularity, the expected value of this number grows like the square root of the corresponding deterministic upper bound as the degree of the polynomial tends to infinity.
Waring problem for forms, which dates back to J.J. Sylvester, has demonstrated to be a fruitful and enriching problem in algebraic geometry, which have found recently wide applications in biology, signal processing, machine learning, quantum information theory... In this talk, we will start overviewing Waring type problems, emphasizing applications.
Thereafter, we will review a couple of results obtained in joint work with Shreedevi K. Masuti for binary forms with complex coefficients. The most remarkable of our results is an explicit formula for the Waring rank of any binomial.
We want to identify and describe the mathematical structure of "sparsity" in high-dimensional problems. Systems that depend on a large number of variables are known to suffer from the curse of dimensionality: their complexity generally grows exponentially in the number of variables. Nevertheless, decades of research have shown that in many cases such systems can be accurately approximated with polynomial complexity. Perhaps the most studied phenomena in this context are entangled quantum mechanical systems obeying Area Laws. In such cases the information content scales much slower than the size of the system.
In this talk we will discuss the links between entropy Area Laws and low-rank approximation. We will see how a local (NNI) operator admits eigenfunctions with favorable approximation properties. This will lead to an Area Law for PDEs. We will conclude by discussing the necessary assumptions, evaluate the potential and limitations of this approach.
A tensor product surface is the image of a rational map p P^1xP^1->P^3. Such surfaces arise in geometric modelling and in this context it is useful to know the implicit equation of the closure of the image. In this talk I will introduce a residual resultant for P^1xP^1 following Gelfand, Kapranov, Zelevinski and show that it can be computed using virtual projective resolutions. Afterwards I will explain how to use the residual resultant in P^1xP^1 to compute implicit equations of tensor product surfaces. Many examples and concrete computations will be presented.
In this talk, we study Lagrangian Fibrations with designed singular fibers. The idea is to construct a $K3$ surface $X$ as a minimal resolution of the singularities of a double cover $Y$ of the plane branched along a reduced but possibly reducible singular sextic $\Sigma$. Moreover, we assume that $\Sigma$ has at worst $A$-$D$-$E$ singularities. This freeness of choosing $\Sigma$ allows us to construct many examples of singular fibres with various singularities.We find an explicit description of the singular fibers of the Lagrangian Fibrations $f\colon M_X(0,2H,\chi)\rightarrow |2H|$. The results shed also some light on the correlation between the degree of the discriminant divisor $\Delta$ and the topology of the corresponding moduli space.
Branched rough paths are a generalization of T. Lyons' geometric rough paths, introduced in 2010 by M. Gubinelli. They have played an important role in the solution theory of singular SPDEs and were one of the sources of inspiration for M. Hairer's theory of Regularity Structures. The problem of existence of a branched rough path above a fixed vector-valued Hölder path is, of course, important. We propose a solution to this problem based on an explicit form of the Baker-Campbell-Hausdorff formula due to Reutenauer, and on the Hairer-Kelly map. We introduce a new class of rough paths which we call anisotropic geometric rough paths. Our techniques also allow us to give an action of a Banach space of Hölder functions on branched rough paths, hence endowing space of branched rough paths with the structure of a principal homogeneous space. (This is a joint work with L. Zambotti)
For a smooth projective variety X with a C^*-action, every orbit compactifies to a map from P^1 and one can split the points of X according to where infinity is mapped. The obtained division is called the negative Bialynicki-Birula decomposition. In the talk I will explain how to generalize the BB decomposition and use it to find new components and prove non-reducedness and other pathologies for the Hilbert scheme of points. Surprisingly, the key role is played by the tangent space to the Hilbert scheme, which is easily computable, hence the search can be effectively conducted. I will also list some open questions.
I will talk about the convex hulls of trajectories of polynomial dynamical systems. Such trajectories also include real algebraic curves. The boundary of the resulting convex bodies are stratified into families of faces. I will discuss the numerical algorithms we developed for identifying these patches of faces. This work is also a step towards computing the attainable region of a trajectory. This is a joint work with Daniel Ciripoi, Andreas Loehne, and Bernd Sturmfels.
Let G be a finite group and let Int(G) be the subgroup of Aut(G) consisting of those automorphisms (called 'intense') that send each subgroup of G to a conjugate. Intense automorphisms arise naturally as solutions to a problem coming from Galois cohomology, still they give rise to a greatly entertaining theory on its own. We will discuss the case of groups of prime power order and see that, if G has prime power order but Int(G) does not, then the structure of G is (surprisingly!) almost completely determined by its nilpotency class. The results I will present are part of my PhD thesis, which I wrote under the supervision of Prof. Hendrik Lenstra at the University of Leiden.
This talk aims to answer the following question: What do we know about the variety defined by the k-signature of an axis path? This is part of the work done during my visit to the MPI together with Mateusz Michalek and Francesco Galuppi.
In recent years, several attempts have been proposed to define a notion of non-commutative toric variety. In this talk, I will report on a work in progress with Francesco Galuppi concerning the definition of a Veronese map between non-commutative algebras. Starting from the definition of the coordinate algebra of the theta-deformed non-commutative projective space, I will describe embeddings of non-commutative projective spaces into the theta-deformed projective line, establishing the necessary conditions for such embeddings to exist and describing the defining ideals for the resulting non-commutative varieties. Finally, I will discuss how this approach relates to existing constructions.
An algebraic manifold is a submanifold of the smooth locus of some real variety embedded in affine space. We can choose a point on it by first choosing a hyperplane of the right codimension and then choosing one of the intersection points of the plane with the manifold. In this talk, I explain how to do this such that the chosen point is uniformly distributed. I show examples of the corresponding algorithm for sampling in action and highlight a connection to topological data analysis.
We consider the problem of finding the isolated points defined by an ideal in a ring of (Laurent) polynomials with complex coefficients. Algebraic approaches for solving this use rewriting techniques modulo the ideal to reduce the problem to a univariate root finding or eigenvalue problem. We introduce a general framework for algebraic solvers in which it is possible to stabilize the computations in finite precision arithmetic. The framework is based on truncated normal forms (TNFs), which generalize Groebner and border bases. The stabilization is based on a `good' choice of basis for the quotient algebra of the ideal and on compactification of the solution space.
A fundamental notion in Tropical Geometry is that of a tropical basis: a set of polynomial equations whose associated tropical variety agrees with the intersection of the associated tropical hypersurfaces. We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide algorithms for finding them around affine spaces of complementary dimension to the zero set. This is joint work with Yue Ren and Jeff Sommars.
We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Amendola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics.
We first recall the well-known regular tree languages, their properties, and several of their characterizations. We then move on to classes of subregular tree languages that are relevant in natural language processing. Despite their rather bad closure properties, the local tree languages are particularly attractive in applications, since they are easy to infer from data. In addition, their expressive power is well understood. A more expressive formalism, called tree substitution grammars, is similarly popular despite its even worse closure properties, but their expressive power is not understood at all. We review the existing results and present the open problems.
We will begin by recalling a well-understood hierarchy of lattice polytope properties. Some of these properties translate into algebraic properties of the corresponding toric ideals. For instance, a polytope possesses a quadratic regular unimodular triangulation if and only if its corresponding toric ideal has a quadratic Groebner basis. The existence of such a triangulation is an open problem for matroid base polytopes, but recently a weaker property was shown resolving a conjecture of Cunningham from 1984.
The condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is important for the stability analysis of algorithms that solve trigonometric mo- ment problems, e.g. Prony’s method. In the univariate case and with well-separated nodes, this condition number is well understood, but if nodes are nearly-colliding, the situation becomes more complicated. After recalling Prony’s method, results for the condition number of Vandermonde matrices with pairs of nearly-colliding nodes are presented. This is joint work with Stefan Kunis.
We study conditional independence of sets of coordinates in a multivariate Gaussian distribution, with an interest in the different conditional independence models that are possible. In one natural case, corresponding to certain memoryless processes that generalise Markov chains, a complete answer can be obtained using the tools of Schubert varieties and determinantal ideals. This is joint work with Jenna Rajchgot and Seth Sullivant.
The Hurwitz space H_(d,g) parametrizes d-sheeted simply branched covers of the projective line by smooth curves of genus g. In this talk, I will survey our knowledge on the unirationality of Hurwitz spaces and we will discuss some results in this direction.
I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, is arbitrarily numerically forward unstable. The analysis shows that this problem is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1× n2× 2 tensors than for the n1× n2× n3 input tensor. Moreover, I present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits. Joint work with Carlos Beltran and Nick Vannieuwenhoven.
In this talk we present some underlying connections between symbolic computation and graph theory. Inspired by the two papers of Cifuentes and Parrilo in 2016 and 2017, we are interested in the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style. Viewing triangular decomposition in top-down style as multivariate generalization of Gaussian elimination, we show that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets, including all the computed triangular sets, appearing in one specific algorithm for triangular decomposition in top-down style (Wang’s method) have associated graphs which are subgraphs of the chordal graph of the input polynomial set. Potential applications of chordal graphs in symbolic computation are also discussed.
To a directed graph one can assign its Leavitt path algebra, which is a quotient of the path algebra of the extended graph by a certain ideal. An appropriate choice of morphisms in a category whose objects are directed graphs makes this assignment into a covariant functor into the category of algebras. In spite of the apparent covariant nature of the construction of Leavitt path algebras, we prove that, for a suitable class of graphs, pushouts of directed graphs give rise to pullbacks of the underlying Leavitt path algebras. This talk is based on the joint work with Piotr M. Hajac and Sarah Reznikoff.
Operator algebras are the language of quantum mechanics just as much as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. One of these endeavours is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C*-algebras) and locally compact Hausdorff spaces. Thus noncommutative C*-algebras are thought of as quantum topological spaces, and are researched from this perspective. However, such C*-algebras can enjoy features impossible for commutative C*-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new ("quantum") intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces, and how it explains its K-theory.
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their signature tensors of order three. We establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. A detailed analysis of the relevant condition numbers is presented. We also compute the shortest path with a given signature tensor.
For matrices with nonnegative entries, the Perron-Frobenius theorem discusses existence, uniqueness, maximality and the computation of eigenvectors with nonnegative entries. This result has been generalized for homogeneous mappings leaving a cone invariant. More recently, Perron-Frobenius type results have been proved in multi-linear algebra, for the study of spectral problems involving tensors with nonnegative entries. We discuss a Perron Frobenius theorem for multi-homogeneous mappings which unify both of these generalizations in a single framework.
In my talk I will propose and discuss a set of combinatorial invariants of simplicial complexes. The invariants are very elementary and defined by counting connected components and/or homological features of induced subcomplexes, but admit a commutative algebra interpretation as weighted sums of graded Betti numbers of the underlying complex.
I will first define these invariants, state an Alexander-Dehn-Sommerville type identity they satisfy, and connect them to natural operations on triangulated manifolds and spheres. Then I will present a (non-optimal) upper bound for arbitrary pure and strongly connected simplicial complexes and discuss the very natural conjecture that, for triangulated spheres of a given f-vector, the invariants are maximised for the Billera-Lee-spheres.
This is joint work with Giulia Codenotti and Francisco Santos, see https://arxiv.org/abs/1808.04220.
Hypertoric variety $Y(A, \alpha)$ is a (holomorphic) symplectic variety, which is defined as a Hamiltonian reduction of complex vector space by torus action. This is an analogue of toric variety. Actually, its geometric properties can be studied through the associated hyperplane arrangements (instead of polytopes). By definition, there exists a projective morphism $\pi:Y(A, \alpha) \to Y(A, 0)$, and for generic $\alpha$, this gives a crepant resolution of affine hypertoric variety $Y(A, 0)$. In general, for a (conical) symplectic variety and its crepant resolution, Namikawa showed the existence of the universal Poisson deformation space of them. We construct the universal Poisson deformation space of hypertoric varieties $Y(A, \alpha)$ and $Y(A, 0)$. We will explain this construction. In application, we can classify affine hypertoric varieties by the associated matroids. If time permits, we will also talk about applications to counting crepant resolutions of affine hypertoric varieties. This talk is based on my master thesis.
I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, is arbitrarily numerically forward unstable. The analysis shows that this problem is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1× n2× 2 tensors than for the n1× n2× n3 input tensor. Moreover, I present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits. Joint work with Carlos Beltran and Nick Vannieuwenhoven.
Solving a (undirected graph) Laplacian system Lx = b has numerous applications in theoretical computer science, machine learning, and network analysis. Recently, the notion of the Laplacian operator L_F for a submodular transformation F was introduced, which can handle undirected graphs, directed graphs, hypergraphs, and joint distributions in a unified manner. In this study, we show that the submodular Laplacian system L_F(x) ∋ b can be solved in polynomial time. Furthermore, we also prove that even when the submodular Laplacian system has no solution, we can solve its regression form in polynomial time. Finally, we discuss potential applications of submodular Laplacian systems in machine learning and network analysis. This is a joint work with Kaito Fujii and Yuichi Yoshida.
Cumulants are basic objects in probability and statistics. In this talk we will discuss the use of certain variation of them to study secant varieties of Segre products. We will further mention some of the geometric properties of these varieties recently investigated by Michalek, Oeding and Zwiernik. Moreover, we will discuss the joint ongoing work with Michalek and Zwiernik about "secant varieties of special embeddings of affine spaces".
The closure of the connected components of the complement of a tropical hypersurface are called regions. They have the structure of convex polyhedra. A 3-dimensional polytope is a K3 polytope if it is the closure of the bounded region of a smooth tropical quartic surface. In this talk we begin by studying properties of K3 polytopes. In particular we exploit their duality to regular unimodular central triangulations of re exive polytopes in the fourth dilation of the standard tetrahedron. Then we focus on quartic surfaces that tropicalize to K3 polytopes, and we look at them through the lenses of Geometric Invariant Theory. This is a joint work with Gabriele Balletti and Bernd Sturmfels.
In this talk I will explain the basic idea of toric degenerations and give an overview on where and how these arise. The motivation is to deform arbitrary (projective) varieties to toric varieties. This is desirable as toric varieties are particularly nice, many of their algebraic and geometric properties are encoded in combinatorial or polyhedral data.
I will explain three approaches (mentioned in the title) to constructing toric degenerations using valuations and show how they are related. The main example will be toric degenerations of Grassmannians, in particular Gr(2,n).
In recent years we have developed a theory that uses Graver bases (certain universal Grobner bases) to solve broad linear and nonlinear integer optimization problems in polynomial time. In the first part of the talk I will overview some of the basics of this theory and its generic application to multiway tables (tensors). In the second part I will describe very recent work that unifies and broadly extends and accelerates this theory.
Probability is a now-classic tool in combinatorics, especially graph theory. Some applications of probabilistic techniques are: (1) describing the typical/expected properties of a class of objects, (2) uncovering phase transitions and sudden thresholds in the dependence of one property on another, and (3) producing examples of extremal or conjectured objects. (This last technique is sometimes called The Probabilistic Method.)
This talk explores these techniques in a commutative algebra setting, using monomial ideals as a bridge between combinatorics and algebra. I'll introduce a family of random models for monomial ideals, and describe the algebraic properties of (quotient rings defined by) these random ideals. We have results of each type described above, for instance: (1) the typical projective dimension of K[x_1,...,x_n] mod a random monomial ideal, (2) thresholds in Krull dimension as a function of number of monomial generators, and (3) how to generate unlimited examples of monomial ideals which aren't generic (in the Bayer-Peeva-Sturmfels sense), but which nevertheless have minimal free resolutions that can be read from their Scarf complexes.
Joint work with Jesus A. De Loera, Sonja Petrovic, Despina Stasi, and Dane Wilburne (pdf: Random Monomial Ideals), De Loera, Serkan Hoşten, and Robert Krone (pdf: Average Behavior of Minimal Free Resolutions of Monomial Ideals), and ongoing work with Hoşten, Wilburne, and Jay Yang.
We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition into simpler (triangular) polynomial sets, while preserving the underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Chordal networks can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, components, and radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.
The uniform probability measure on a convex polytope induces piecewise polynomial densities on the projections of that polytope. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated, and they offer nice challenges for both numerical and symbolic computing in algebraic geometry. This is joint work with Kathlen Kohn and Boris Shapiro.
Persistent homology is a tool used to analyse topological features of data. In this talk, I will describe a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves high performance on several classification benchmarks.
We give examples of smooth plane quartics over Q with complex multiplication over \bar{Q} by a maximal order with primitive CM type. We describe the required algorithms as we go: these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.
An elliptic ruled surface is a 4-manifold satisfying the condition that it has a holomorphic fibration over an elliptic curve with fibers that are projective lines. Every elliptic ruled surface is algebraic, and, in particular, a Kaehler surface. In this talk I would like to discuss the symplectomorphism group of elliptic ruled surfaces. More precisely, we will show that every symplectomorphism that is smoothly isotopic to the identity is isotopic to the identity within the symplectomorphism group.
The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the “effective number of points” of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. In their paper Leinster and Shulman list a series of open questions, two of which are as follows:
1) Magnitude homology only “notices” whether the triangle inequality is a strict equality or not. Is there a “blurred” version that notices “approximate equalities”? 2) Almost everyone who encounters both magnitude homology and persistent homology feels that there should be some relationship between them. What is it?
In this talk I will introduce magnitude and magnitude homology, answer these two questions and show that they are intertwined: it is the blurred version of magnitude homology that is related to persistent homology. Leinster and Shulman's paper can be found at https://arxiv.org/abs/1711.00802.
Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. This talk focuses on certain biological signaling networks, namely, multisite phosphorylation networks. Many of these systems exhibit “toric steady states” (that is, the ODEs generate a binomial ideal), and more generally the set of steady states admits a rational parametrization, that is, the set is the image of a map with rational-function coordinates. We describe how this parametrization allows us to investigate the dynamics of two multisite phosphorylation networks: the emergence of bistability in a network underlying ERK regulation, and the capacity for oscillations in a mixed processive/distributive phosphorylation network. This is joint work with Carsten Conradi and Maya Mincheva.
Computer scientists have conjectured that it is nearly as easy to multiply large matrices as it is to add them. They define the exponent of matrix multiplication $\omega$ to be the infimum of the numbers $\tau$ such that $n\times n$ matrices may be multiplied using $O(n^{\tau})$ arithmetic operations. The conjecture is that $\omega =2$. The problem was posed in 1969 and there was steady progress on proving upper bounds for $\omega$ that ended in 1989. As a first attempt to unblock research on the exponent, I will discuss variants of the conjecture from a geometric perspective. Independent of matrix multiplication, it leads to new, previously uninvestigated properties of tensors of interest in their own right. This is joint work with A. Conner, F. Gesmundo, E. Ventura and Y. Wang.
At the conference FPSAC 2017 in London, Luc Lapointe gave a plenary talk about his work, together with many other collaborators, about superspace. After attending the talk, Mike Zabrocki and I were very interested in this work and we start reading about it and having questions (and conjectures!). In this talk, I would like to present what it is known about superspace, what I am interested in and other open questions.
Automated invariant generation is a fundamental challenge in program analysis and verification, with work going back several decades. In this talk I will present a select overview and survey of previous work on this problem, and I will describe a new algorithm to compute all polynomial invariants for the class of so-called affine programs (programs that allow affine assignments and non-deterministic branching). Our main technical contribution is a mathematical result of independent interest: an algorithm to compute the Zariski closure of a finitely generated matrix semigroup.
This is joint work with Ehud Hrushovski, Joel Ouaknine, and Amaury Pouly.
A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. A set of generators of the fundamental group for the complement of the critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw–Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw–Curtis algorithm and contour integrals. Siegel's algorrithm is applied to approximately identify a symplectic transformation to the fundamental domain in order to allow for an efficient computation of multi-dimensional theta functions.
We will describe an algorithm to compute the period matrix of projective hypersurfaces, in particular, of plane curves. This algorithm computes the value of the period integrals without taking any Riemann integrals but by solving an initial value problem for a system of ordinary differential equations. With this approach we bypass the need to determine explicit homology cycles on the hypersurfaces, which is the key to period computation beyond dimension one. Furthermore, as quadrature is avoided, periods can be computed to extreme-precision.
We present a short review comparing Baker-Akhiezer and secant identity approaches of the appearence of multidimensional theta functions in the theory of integrable partial differential equations. As examples we discuss the Kadomtsev-Petviashvili equation, a two-dimension generalization of the celebrated Korteweg-de Vries equation, and the Ernst equation which is equivalent to the stationary axisymmetric Einstein equations in vacuum. Solutions to the latter are given on a family of hyperelliptic curves which makes an efficient numerical treatment of modular functions associated to such curves necessary in order to discuss these solutions. We present an efficient numerical approach to hyperelliptic curves based on Clenshaw-Curtis integration which computes all needed quantities even in almost degenerate situations to machine precision.
In this talk I will explain 1) how to construct a very simple moduli space (=a projective space) for the non-decomposable P1-bundles with no "jumping fibres" over Hirzebruch surfaces, and 2) how to deduce the classification of all the P1-bundles over rational surfaces whose automorphism group G is maximal in the sense that every G-equivariant birational map to another P1-bundle is necessarily a G-isomorphism. (This is a joint work with Jérémy Blanc and Andrea Fanelli.)
We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.
We will discuss the spectral theory of the locally symmetric varieties associated to reductive groups defined over number fields. A fundamental tool is Arthur's trace formula, which in a first approximation expresses the duality between the spectrum and the conjugacy classes of the group of rational points. However, the picture is complicated by the contribution of the continuous spectrum to the spectral side and by the terms corresponding to non-elliptic conjugacy classes on the geometric side. We will review the absolute convergence and continuity of the trace formula for a large natural class of test functions. We will then discuss a concrete application to the asymptotics of traces of Hecke operators (for classical groups and G2). This is joint work with Erez Lapid, Werner Mueller and Jasmin Matz.
Motivated by the problem of intrinsic tropicalization of very affine varieties, we compute the group of units of the coordinate rings of smooth curves of low genus, providing explicit algorithms to do so in each case. Our techniques involve interpolation in the rational case, and methods from algebraic number theory in the elliptic case.
(joint work with Leon Zhang and Justin Chen)
We study the regularity for finite modular lattices which are not distributive. In particular, we show that the join-meet ideal of such a lattice does not have a linear resolution. Joint work with V. Ene and T. Hibi.
The Stanley-Reisner correspondence, which assigns a commutative ring to each finite simplicial complex, is a useful and well-studied bridge between commutative algebra and combinatorics. In 1987 Sergey Yuzvinsky proposed a construction that allows to see the Stanley-Reisner ring of a finite simplicial complex as the ring of global sections of a sheaf of rings on a poset. Motivated by applications in the theory of Abelian arrangements, Emanuele Delucchi and I extend Yuzvinsky's construction to the case of (possibly infinite) finite-length simplicial posets. This generalization behaves well with respect to quotients of simplicial complexes and posets by translative group actions.
Maximum entropy probability distributions are important for information theory and relate directly to exponential families in statistics. Having the property of maximizing entropy can be used to define a discrete analogue of the classical continuous Gaussian distribution. We present a parametrization of such a density using the Riemann Theta function, use it to derive fundamental properties and exhibit strong connections to the study of abelian varieties in algebraic geometry. This is joint work with Carlos Améndola (TU Munich).
Staged trees are a new and exciting class of statistical models that generalise the well known Bayesian Networks. In this talk I will explain the algebraic and statistical properties of these models and characterise the case in which they are toric varieties.
We introduce a class of graphs called trimmable. Then we show that the Leavitt path algebra of a trimmable graph is graded-isomorphic to a pullback algebra of simpler Leavitt path algebras and their tensor products.Next, specializing the ground field to the field of complex numbers and completing Leavitt path algebras to graph C*-algbras, we prove that the graph C*-algebra of a trimmable graph is U(1)-equivariantly isomorphic with an appropriate pullback C*-algebra.As a main application, we consider a trimmable graph yielding the C*-algebra $C(S^{2n+1}_q)$ of the Vaksman-Soibelman quantum sphere, and use the resulting pullback structure of its gauge invariant subalgebra $C(CP^n_q)$ defining the quantum complex projective space to show that the generators of the even K-group of $C(CP^n_q)$ are given by a Milnor connecting homomorphism applied to the (unique up to sign) generator of the odd K-group of $C(S^{2n-1}_q)$ and by the generators of the even K-group of $C(CP^{n-1}_q)$. Based on joint works with Francesco D'Andrea, Atabey Kaygun and Mariusz Tobolski.
A homogeneous polynomial of degree d in n+1 variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in n+1 variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces. This is a joint work with Massimiliano Mella.
Generalized exponents are important graded multiplicities in representation theory of simple Lie algebras. Notably, they are particular Kazhdan-Lusztig polynomials. In type A, they admit a nice combinatorial description in terms of Lascoux-Schützenberger’s charge statistics on semistandard tableaux. In this talk I will recall their definition and explain how to get similar statistics beyond type A. This will give a combinatorial proof of the positivity of their coefficients but also some other interesting properties. This is a work in collaboration with Cristian Lenart.
Recently several conjectures were resolved by using a classical method for constructing vectors in irreducible representations of coordinate rings of group varieties. We explain the method and several applications, for example a proof of Weintraub's 1990 conjecture on plethysm coefficients (with Bürgisser and Christandl) and a disproof of Mulmuley and Sohoni's 2008 occurrence obstruction conjecture in geometric complexity theory (with Bürgisser and Panova). Moreover, we explain how we were able to shed more light on several other conjectures by running the method on cluster computers.
Yield stress fluid flows play an important role in the oil and gas industry. There are many numerical methods for modeling such flows, and they often are computationally expensive. We propose to reduce the computational complexity of parametric studies of Bingham fluid flows by utilizing machine learning techniques. The idea is as follows: instead of solving the PDE for each parameter value, we first do several simulations for a few scenarios (build a training dataset), construct a surrogate model to predict the solution for any parameter value. In this case, we have a much faster model. We apply this approach to a well-known Mosolov problem with Bingham number as a parameter.
Deep neural networks and tensors are different forms of approximation of multivariate functions. In this talk, I will give an overview of our recent results on tensor and matrix analysis, deep learning, and their connections
1) Desingularization of low-rank matrix manifolds (joint with V. Khrulkov)
2) The expressive power of recurrent neural networks (joint with V. Khrulkov and A. Novikov)
3) Universal adversarial examples and singular vectors (joint with V. Khrulkov)
4) Geometry score: a way to compare generative adversarial networks (joint with V. Khrulkov)
The development of efficient numerical algorithms for solving large-scale linear systems is one of the success stories of numerical linear algebra that has had a tremendous impact on our ability to perform complex numerical simulations and large-scale statistical computations. Many of these developments are based on multilevel and domain decomposition techniques, which are intimately linked to Schur complements and low-rank updates of matrices. These tools do not carry over in a direct manner to other important linear algebra problems. Two such problems are the computation of matrix functions and the solution of matrix equations. They arise in a wide variety of application areas, including control, signal processing, network analysis, and computational statistics. Despite impressive progress made during the last decades on both problems, there remain numerous situations in which existing algorithms become computationally too expensive. This includes seemingly simple tasks, such as computing the diagonal of a matrix function for a very large sparse matrix.
In this talk, we describe a new framework for performing low-rank updates of matrix functions. This allows to address a wide variety of matrix functions and matrix structures, including sparse matrices as well as matrices with hierarchical low rank and Toeplitz-like structures. The framework is quite versatile and can be adaptated to seemingly unrelated problems, such as computing and updating determinants of large-scale matrices.
The talk is based on joint work with Bernhard Beckermann and Marcel Schweitzer.
Interlacing polynomials are a powerful method to prove that a polynomial has only real roots. In this talk I present applications to h*-polynomials of dilated lattice polytopes.
Number fields, which are finite field extensions of the field of rational numbers, are fundamental objects of study in number theory. One of the most important invariants of a number field is the class group, which measures how close the ring of integers of that number field is to being a unique factorization domain. Despite their importance, class groups remain mysterious objects in number theory. This talk will demonstrate how the titular family of curves can be used to construct cubic number fields with interesting class groups.
Background: The talk is designed for a general level mathematics colloquium. Some algebraic number theory or algebraic geometry is useful, but not necessary.
Inverse spectral geometry studies in what extent the spectrum of the Laplace operator determines the geometry of a Riemannian manifold. The interest on this area increased a lot after M. Kac's article "Can one hear the shape of a drum?" in the 60's.
It has been recently discovered that the spectrum of the Laplace operator of a lens space (a quotient of a sphere by a cyclic group) can be encoded by the Ehrhart series of certain (very particular) polytope. It may be expected that some problems in spectral geometry can be solved by using Ehrhart theory.
In this talk, we will recall the mentioned connection in an elementary way. It will not be assumed any knowledge on spectral geometry.
In their beautiful paper, Gruson, Lazarsfeld and Peskine prove that if C ⊂ Pr is a projective curve of degree d ≥ r + 2 then its Castelnuovo-Mumford regularity reg(C) is at most d − r + 2 and the equality is attained exactly when C admits a (d−r +2)-secant line. In this talk I will speak about a few generalizations of Gruson-Lazarsfeld-Peskine’s work.
When we study algorithmic or constructive Resolution of Singularities, we make use of invariants that allow us to distinguish among different singular points of an algebraic variety. Attending to them, we choose the centers of a sequence of blow ups that will eventually lead to a resolution of the singularities of the initial variety.
On the other hand, arc spaces are useful in the study of singularities, since they detect properties of algebraic varieties, including smoothness. They also let us define numerous invariants. In particular, the Nash multiplicity sequence is a non-increasing sequence of positive integers attached to an arc in the variety which stratifies the arc space. As we will see, this sequence gives rise to a series of invariants of singularities which turn out to be strongly related to those that we use for constructive resolution of singularities for varieties defined over fields of characteristic zero. Moreover, these invariants defined by means of arc spaces do not rely on the peculiarities of the characteristic zero case, so they pose interesting questions for the case of varieties defined over perfect fields, regardless of their characteristic.
Given a point in a polytope, what is the smallest face of the polytope that contains this point? Even though this problem can be formulated as a linear program, there are cases in applications where the answer cannot be found efficiently. I present methods to solve this problem approximately. The idea is to describe faces by the vertices that lie on this face. This set of vertices can be approximated by a superset or subset using projections or liftings of the polytope.
Motivation for this work comes from discrete graphical models. For such models, the question whether the MLE exists can be reformulated as the question whether the empirical distribution lies on a proper face. Any information about the face of the empirical distribution can help to determine which of the parameters of the graphical model can be estimated reliably.
Given a point in a polytope, what is the smallest face of the polytope that contains this point? Even though this problem can be formulated as a linear program, there are cases in applications where the answer cannot be found efficiently. I present methods to solve this problem approximately. The idea is to describe faces by the vertices that lie on this face. This set of vertices can be approximated by a superset or subset using projections or liftings of the polytope.
Motivation for this work comes from discrete graphical models. For such models, the question whether the MLE exists can be reformulated as the question whether the empirical distribution lies on a proper face. Any information about the face of the empirical distribution can help to determine which of the parameters of the graphical model can be estimated reliably.
In this talk, we study the minimal free resolution of divisors on a smooth rational normal surface scroll $S=S(a_1 ,a_2 )\subset \P^r$. Our main result provides a nice decomposition of the Betti tables of divisors into much simple Betti tables under the suitable conditions.
The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which refines this bound for a particular class of real algebraic projective hypersurfaces in terms of the Hodge numbers of its complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg, we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalisation. Lurking in the spectral sequences of the proof are the keys to controlling the topology of the real hypersurface produced from a patchwork. This is joint work in preparation with Arthur Renaudineau.
A flat surface is a polygon in the plane where we have identified parallel sides or equivalently a complex curve together with a choice of a holomorphic differential form. Flat surfaces with many real symmetries give rise to special curves inside the moduli space of curves called Teichmüller curves. Flat surfaces with extra holomorphic symmetries correspond to orbifold points on such Teichmüller curves. We provide a topological classification of some Teichmüller curves by explicitly counting their orbifold points.
Some of the oldest problems in mathematics amount to finding solutions in rational numbers to equations describing curves. Faltings established the landmark result that curves of general type have only finitely many rational points, but his method of proof is ineffective. The problem of deciding if an arbitrary curve has any rational points at all remains unsolved.
I will discuss some of the methods that can, under some conditions, decide if a curve has no rational points, and determine the rational points otherwise. While we know that for each of these methods there exist curves for which the methods fail, we know thanks to work by Bhargava et al. that at least for hyperelliptic curves the methods do work for the majority of them.
In this talk we present connections between the true persistent homology of algebraic varieties and their corresponding offset hypersurfaces. By Hardt's theorem, true persistent homology events occur at algebraic numbers. By studying the offset hypersurface, we characterize when true persistent homology events occur. The degree of the offset hypersurface is bounded by the Euclidean Distance Degree of the starting variety, so this way we show a bound on the degree of the algebraic numbers at which the homological events occur.
The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. The talk will not assume any prior knowledge of oriented matroids.
Principal component analysis is a widely-used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes. This is joint with with Leon Zhang and Xu Zhang.
One of the most important invariants of a lattice polytope is the Ehrhart polynomial encoding the number of lattice points contained in its integral dilation. In this talk, after surveying the Ehrhart polynomials of lattice polytopes and their fundamental properties, we will focus on the characterization problem on the Ehrhart polynomials and give some recent results.
Classical algorithms for numerical integration (quadrature/cubature) proceed by approximating the integrand with a simple function (e.g. a polynomial), and integrate the approximant exactly. In high-dimensional integration, such methods quickly become infeasible due to the curse of dimensionality.
A common alternative is the Monte Carlo method (MC), which simply takes the average of random samples, improving the estimate as more and more samples are taken. The main issue with MC is its slow "sqrt(variance/#samples)" convergence, and various techniques have been proposed to reduce the variance.
In this work we reveal a numerical analyst's interpretation of MC: it approximates the integrand with a simple(st) function, and integrates that function exactly. This observation leads naturally to MC-like methods that combines MC with function approximation theory, including polynomial approximation, sparse grids and low-rank approximation. The resulting method can be regarded as another variance reduction technique for Monte Carlo.
A T-design is a collection of subsets of a set X of size k, called "blocks", such that every subset of X of size T is contained in lambda blocks. Examples of designs are magic and latin squares, Hadamard matrices, Steiner systems, finite projective planes, to name just a few.
From a T-design I will show how to construct a T-dimensional tropical variety of degree lambda. Then using this construction, I will interpret some operations on designs through an algebro-geometric lens. This work is very much in progress and comments/suggestions are welcome.
In this talk we will present a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic.This formula also provides a new method for evaluating and computing this discriminant more efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we derive new properties and we show that this new definition of the discriminant satifises to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. More precisely, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Matroids over hyperfields, introduced by Baker and Bowler in 2016, offer a new unifying vision encompassing matroids, vector spaces, and related ideas in tropical geometry. The key tool are hyperfields, i.e., algebraic structures akin to fields — but with multivalued addition.
In this talk I will define matroids over hyperfields, present some examples (especially as related to tropical geometry and oriented matroids) and I will discuss some of the first results in this very young subject (which can be understood as looking for a “linear algebra” over hyperfields). The exposition will not presuppose any special previous knowledge.
Sensitivity analysis in probabilistic discrete graphical models is usually conducted by varying one probability value at a time and observing how this affects output probabilities of interest. When one probability is varied then others are proportionally covaried to respect the sum-to-one condition of probability laws. The choice of proportional covariation is justified by a variety of optimality conditions, under which the original and the varied distributions are as close as possible under different measures of closeness. For variations of more than one parameter at a time proportional covariation is justified only in some special cases. In this work, for the large class of models entertaining a monomial parametrisation, we demonstrate the optimality of newly defined proportional multi-way schemes with respect to an optimality criterion based on the notion of I-divergence. Furthermore we introduce a condition that any proportional covariation needs to respect in order to be optimal. This is shown by adopting a new formal, geometric characterization of sensitivity analysis in monomial models, which include a wide array of probabilistic graphical models. We also demonstrate the optimality of proportional covariation for multi-way analyses in Naive Bayes classifiers.
This is joint work with Manuele Leonelli, University of Glasgow, UK
The integer linear programming approach to structural learning graphical models is based on the idea to represent them by means of special vectors, whose components are integers. In the context of learning decomposable models, we propose to represent them by special zero-one vectors, named characteristic imsets (of the corresponding Bayesian network model) [1]. This idea leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs we name the chordal graph polytope [2]. The talk will be devoted to the attempts to characterize theoretically all facet-defining inequalities for this polytope in order to utilize that in ILP-based procedures for learning decomposable models.
The talk is based on joint research with James Cussens from York University, UK.
Persistent homology (PH) is one of the most successful methods in topological data analysis, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance. PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multiparameter PH is hard, and it presents one of the biggest challenges of topological data analysis.
In this talk I will introduce persistent homology, give a brief overview of the complexity of the theory in the multiparameter case, and then discuss how tools from commutative algebra give invariants suitable for the study of data.
This is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann.
For every finite reflection group W, the noncrossing partitions NC(W) are a lattice in the absolutely ordered group W. In the case of the symmetric group, T. Brady and J. McCammond showed that the order complex of NC(W) embeds into a spherical building. In joint work with P. Schwer, we generalized the theorem for all finite reflection groups. Interpreting NC(W) as a subcomplex of a spherical building now enables us to better understand the building-like structure of NC(W).
Numerical algebraic geometry uses numerical algorithms to study algebraic varieties, which are sets defined by polynomial equations.
It is becoming a core tool in applications of algebraic geometry outside of mathematics. Its fundamental concept is a witness set which gives a representation of a variety that may be manipulated on a computer. A trace test is used to verify that a witness set is complete and provide stopping criteria for algorithms. Recently, there have been further developments in computational algebraic geometry to study varieties in a product space, which is the natural setting for solutions of parameterized polynomial systems. In this talk, we discuss some of these developments and a new trace test for these varieties. We demonstrate the methods' effectiveness in examples from various areas of science and statistics.
In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. (Bio)chemical reaction networks define systems of ordinary differential equations with (in general, unknown) parameters. Under mass-action kinetics, these equations depend polynomially on the concentrations of the chemical species. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need a priori determination of the parameters, which can be theoretically or practically impossible.
I will first present a gentle introduction to the basic concepts. I will then describe general results based on the network structure. In particular, I will explain a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many post-translational modification networks. I will also outline recent methods to detect the capacity for multistationarity and to describe parameters for which multistationarity occurs, allowing for multiple steady states with the same total amounts.
We define a geometric RSK correspondence for any semisimple group and any reduced decompositon of element of its Weyl group. This correspondence is a biration map of tori of dimension equal to the length of a reduced decomposition.For the longest element of the Weyl group, the tropicalization of this map turns out to be the isomorphism between the Lusztig crystal on the canonical basis and the Kashiwara crystal on the dual canonical basis. The geometric corespondence provide us with a transformation of the corresponding superpotentials for geometric crystals. For the case $G=SL_{n+m}$ and the grassmannian permutation, our construction lead to a modified variant of usual RSK being a bijection between non-negative $n\times m$ arrays and pairs of semistandard Young tableaux of equal form (modifiction means that we get a pair $(P^{Sch}, Q)$ for an array, where $P^{Sch}$ denotes the tableaux being the Sch\"utzenberger involution to $P$, while usual RSK ends with the pair $(P,Q)$). The geometric RSK, in such a case, transforms the Berenstein-Kazdan potential to the superpotential due to Riesch-Williams.
The unipotent radical U of a Borel subgroup of SL_n(C) posses the structure of a Cluster Variety. Gross-Hacking-Keel-Kontsevich constructed a canonical basis B of the coordinate ring C[U]. The basis B is parametrized by a cone K in the tropical points of a dual cluster variety. In this talk we introduce certain operators on K generating the cone recursively. As a consequence we obtain a crystal structure on B in the sense of Kashiwara.
Deciding wether a 2D rod-and-pinion framework is rigid can be done by checking that its underlying graph satisfies the Laman conditions. For frameworks with a special configuration such as grids of squares, there is a simpler way to associate a graph to the framework and decide if it is rigid or not. In this talk I will consider frameworks that come from Penrose tilings and show that we can decide the rigidity of these graphs as we do for grids of squares. There is no generalization of Laman conditions for rigidity of 3D graphs but perhaps we can prove a generalization of 2D results for cubical frameworks. Pictures and real time interactive animations will be present throughout this talk to illustrate important concepts.
The class of split matroids arises by putting conditions on the system of split hyperplanes of the matroid base polytope. It can alternatively be defined in terms of structural properties of the matroid. We use this structural description to give an excluded minor characterisation of the class. A small introduction to matroid theory will be given before motivating and explaining the result.
In convex algebraic geometry, a problem that has generated a lot of interest is the problem of determinantal representations of polynomials and these determinantal polynomials play a crucial role in semidefinite programming problems . In my talk, we discuss the problem of representing a multivariate real polynomial of (total) degree d as the determinant of a monic linear matrix polynomial whose coefficient matrices are either symmetric/ Hermitian matrices of order d. In particular, I shall talk about a complete characterization of quadratic determinantal polynomials and propose a method to compute a monic symmetric/Hermitian determinatal representation for a bivariate polynomial if it exists using the theory of majorization and exterior algebra.
Gaussoids offer a new link between combinatorics, statistics and algebraic geometry. Introduced by Lnenicka and Matus in 2007, their axioms describe conditional independence for Gaussian random variables. This lecture introduces gaussoids to an audience familiar with matroids. The role of the Grassmannian for matroids is now played by a projection of the Lagrangian Grassmannian. We discuss the classification and realizability of gaussoids, and we explore oriented gaussoids, valuated gaussoids, and the analogue to positroids. This is based on recent work with Tobias Boege, Alessio D'Ali, and Thomas Kahle.
In most tensor spaces, the sharp estimate for the ratio between spectral and Frobenius norm is not known. However, using an adequate generalization of orthogonality, we can show that in those spaces in which orthogonal tensors exist, the extremal ratio can be determined and is attained only for multiples of such tensors. The existence of a real orthogonal third-order tensor is equivalent to a composition formula for bilinear forms (Hurwitz problem). We also present an inductive construction for certain tensors of order higher than three. Interestingly, orthogonal tensors (in the proposed sense) do not exist in higher-order complex spaces.
I will introduce prolongations and envelopes of a family of submanifolds or subvarieties, in the generality of arbitrary dimensions and jet-order. The notion has its roots throughout classical geometry (optics, caustics, developable surfaces, confocal quadrics).
I will define coisotropic hypersurfaces as in the book by Gel'fand, Kapranov and Zelevinsky. Subsequently, we will discuss coisotropic varieties as their generalization and see first non-trivial examples for this new notion.
In 1980 White conjectured that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. Herzog and Hibi go even further - they ask if the toric ideal of a matroid possesses a Grobner basis of degree 2. We study these problems for a class of matroids of fixed rank, and obtain several finiteness results.
We prove White's conjecture for `high degrees'. That is, we prove that for all matroids of fixed rank r, homogeneous parts of degree at least c(r) of the corresponding toric ideals are generated by quadratic binomials corresponding to symmetric exchanges. This extends our previous result (with Mateusz Michalek) confirming the conjecture `up to saturation'. We also prove that for the class of matroids of fixed rank, there exists a common upper bound on the degree of a Grobner basis. Namely, we prove that the toric ideal of a matroid of rank r possesses a Grobner basis of degree at most 2(r + 3)!.
We focus on geometric (convex and algebraic) aspects of the positive semidefinite matrix completion problem. Of particular interest will be the geometry of Hankel spectrahedra.
Given a real algebraic curve we consider the set of all morphisms to the projective line with the property that the preimage of every real point consists entirely of real points. It turns out that this generalises the notion of interlacing polynomials on the real line to projective curves. Using this theory, we will answer a question raised by Shamovich and Vinnikov on hyperbolic curves as well as a question by Fiedler-LeTouzé on totally real pencils on plane curves. This is joint work with Kristin Shaw.
Local cohomology was introduced by Grothendieck in early 1960s. Since then it has become an essential tool and active research topic in commutative algebra and algebraic geometry. While local cohomology modules contain many useful and subtle information, one fundamental obstacle in understanding them is that they are often not finitely generated. Let I be a graded ideal in a polynomial ring over a field. In this talk I will describe a recent work with Jonathan Montano on estimating the Hilbert functions of the local cohomologies of powers of I. One particularly interesting and crucial case is when I is a monomial ideal, where it turns out that the counting function for the graded pieces can be described using Presburger arithmetic.
Studying regular subdivisions of a class of convex polytopes which are called hypersimplices naturally leads to a new class of matroids, which we call split matroids. Very many matroids are of this type; e.g., the paving matroids arise as special cases. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry. Joint work with Benjamin Schröter.
In this talk I will present joint work in preparation with Philipp Jell and Johannes Rau on a tropical version of the Lefschetz (1, 1) theorem.
In complex algebraic geometry this theorem completely describes which degree 2 cohomology classes of a smooth projective variety can be represented by (Poincare duals to) algebraic cycles. For tropical varieties, we of course must consider tropical cohomology classes. This is just the cohomology of certain cellular sheaves on a tropical variety and can be computed using the cellularSheaves package for polymake (joint work with Lars Kastner and Anna-Lena Winz).
The tropical analogue of the Lefschetz (1,1) theorem uses a "wave map” on cohomology introduced by Mikhalkin and Zharkov. I will explain how this map has the ability to detect tropical algebraic cycles and how it might also be used in moduli problems, for example in the case of K3 surfaces.
An important task in computer vision is the reconstruction of a 3D scene from multiple pictures. The core of this problem requires solving a system of polynomial equations, which leads to interesting problems in projective and algebraic geometry.In this talk, I will present a new general framework for 3D vision, in which a camera is modeled geometrically as a mapping from $P^3$ to a line congruence. This viewpoint applies to traditional pinhole cameras, but also to many other practical devices, such as two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. The multi-view geometry of all these systems can be studied using the "concurrent lines variety", which consists of n-tuples of lines in $P^3$ that intersect at a point. Moreover, several classical features of pinhole cameras, such as intrinsic and extrinsic parameters, can be defined in a very general setting.
A cycle of elliptic curves consists of elliptic curves defined over finite fields, such that the number of points on one curve is equal to the size of the field of definition of another, in a cyclic way. Such cycles are known to exist for arbitrary lengths, but their properties are not well studied. We discuss constructions of cycles with small embedding degrees, as well as their applications to cryptography. We also formulate open problems about these cycles.
Algebraic methods for optimization have been used for maximum likelihood estimation and optimizing Euclidean distance. A complexity measure of these problems is given by the algebraic degree of the problem. This is a good measure because it counts the number of trials needed to solve the problem. These optimization problems are described by a correspondence between critical points, Lagrange-multipliers, and functions to be optimized. In this talk we present a study of subfamilies of these problems. Our main contribution is to formulate a unified duality theory encompassing maximum likelihood degree and Euclidean distance degree duality. We use this duality theory to develop algorithms and describe special loci. Moreover, we do a case analysis for problems in statistics, kinematics, and engineering.
The multiview variety associated to a collection of N cameras records which sequences of image points in P^2N can be obtained by taking pictures of a given world point x∈P3 with the cameras. In order to reconstruct a scene from its picture under the different cameras it is important to be able to find the critical points of the function which measures the distance between a general point u∈P^2N and the multiview variety. We calculate a specific degree 3 polynomial that computes the number of critical points as a function of N. In order to do this, we construct a resolution of the multiview variety, and use it to compute its Chern-Mather class.
A data set is often given as a point cloud, i.e. a non-empty finite metric space. An important problem is to detect the topological shape of data — for example, to approximate a point cloud by a low-dimensional non-linear subspace such as a graph or a simplicial complex. Classical clustering methods and principal component analysis work very well when data points split into well-separated groups or lie near linear subspaces.
Methods from topological data analysis detect more complicated patterns such as holes and voids that persist for a long time in a 1- parameter family of shapes associated to a point cloud. These features were recently visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimal spanning tree of a point cloud to a graph with cycles. I will talk about a generalization of this 1-skeleton to higher dimensions and optimality results that we proved.
The study of resolutions and syzygies of modules has been a topic of interest for commutative algebraists and algebraic geometers since the work of David Hilbert and, more recently, the inception of homological algebra.
The aim of this talk is twofold. The first half will serve as a very brief introduction to the subject, with a focus on its combinatorial aspects. As an application, in the second half we will present some results on some ideals associated with posets (joint work with Gunnar Fløystad and Amin Nematbakhsh), giving a common framework for many well-studied classes of ideals. The techniques used to prove these results are both topological-combinatorial and algebraic.
The double ramification cycle on the moduli space of marked curves of compact type parametrizes marked, stable, compact type curves admitting a finite degree covering of the projective line with prescribed ramification over zero and infinity. In this talk I will discuss a comparison theorem over the compact type locus between double ramification cycles and classes pushed forward from spaces of rubber stable maps to the moduli space of curves. Recent generalizations of this whole situation will also be discussed. This is joint work with Jonathan Wise and Gabriele Mondello.
Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In my talk various reflexive polytopes arising from perfect graphs will be discussed. No special knowledge will be required to understand my talk.
Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.
The talk will present the concept of a protein molecule and its binding polynomial, and present the problem of decoupling molecules in terms of solving certain systems of polynomial equations. We present some calculations of the mixed volumes associated, using diverse methods, and present a conjecture on the number of solutions (which represent the number of "decoupled" molecules).
Polyhedral methods have been used for many years in social choice where they can be applied to compute the probabilities (and even exact numbers) of election results that yield unexpected phenomena, for example the famous Condorcet paradox. So far such computations have almost exclusively been restricted to elections with three candidates. Now our software Normaliz can compute many such probabilities also for four candidates. We will explain the method, discuss some examples and sketch the algorithmic approach.
There has been massive efforts to understand the parameter spaces of convex polytopes -- and great results such as the “g-Theorem” were achieved on the way. On the other hand, key questions are still open, already and in particular for the case of 4-dimensional polytopes/3-dimensional spheres. One crucial question is “fatness problem” for 4-dimensional polytopes, which is a key to the question whether we should expect the same answers for convex polytopes (which are discrete-geometric objects) and for cellular spheres (a topological model), at least asymptotically.
I will argue that we should not, and present first results in this direction: The sets of f-vectors of 3-spheres and of 4-polytopes do not coincide. One of the efficient key tools in the computations for that is the “biquadratic final polynomials” which may be derived from a linearization of the non-linear (quadratic) Grassmann-Plücker relations. How about inscribable polytopes - to add another non-linear condition?
(Joint work with Philip Brinkmann and others)
Any non-hyperelliptic curve of genus g admits an embedding into the projective space of dimension g-1 via its space of holomorphic differentials. This is the only embedding that can be consistently defined over every family of non-hyperelliptic curves and is deservedly called the canonical embedding.The general curve of genus g, canonically embedded in $P^{g-1}$, will admit exactly $2^{g-1}*(2^g - 1)$ hyperplanes tangential to the curve at g-1 points. The configuration of these hyperplanes and their points of tangency admit numerous geometric questions, most of which are essentially impossible to answer directly.We will demonstrate how such questions can be answered by degenerating a smooth curve to a nodal curve and by following these points of tangency all the way to the nodal curve. Theta characteristics provide us with a convenient language and the means to study infinitesimal perturbations of such degenerations.
In algebraic statistics, Jukes-Cantor and Kimura models are of great importance. Sturmfels and Sullivant generalized these models associating to any finite abelian group a family of toric varieties. In this talk, we will sketch how to prove that the ideals of these toric varieties are generated in bounded degree (which only depends on the group). For the Kimura 3-parameter model, we show that these ideals are generated in degree at most four.
We show how to construct a $C^2$ multiscale approximation scheme for functions defined on the (Riemann) sphere. Based on a 3-directional box-spline, a flexible $C^2$ subdivision scheme over a valence 3 extraordinary vertex can be constructed. We will explain this in detail. This subdivision scheme can be used to model spherical surfaces based on a recursively subdivided tetrahedron, with only valence 3 and 6 vertices in the resulted triangulations. This adds to the toolbox of subdivision methods a high order, high regularity scheme which can be beneficial to scientific computing applications. For instance, the scheme can be used in the numerical solution of the Canham-Helfrich-Evans models for spherical and toroidal biomembranes. Moreover, the characteristic maps of the subdivision scheme endow the underlying simplicial complex with a conformal structure. This in particular means that the special subdivision surfaces constructed here comes with a well-defined harmonic energy functional, which can in turn be exploited to promote conformality in surface parameterizations. We develop an effifficient parallel algorithm for computing the harmonic energy and its gradient with respect to the control vertices.
Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Typically, the occurring grain cells have the combinatorial types of 3-dimensional simple polytopes, and together they tile 3-dimensional space.
We will see that some of the occurring grain types are substantially more frequent than others - where the frequent types turn out to be "combinatorially round". Here, the classification of grain types gives us, as an application of combinatorial low-dimensional topology, a new starting point for a topological microstructure analysis of steel.
Parameterized curves and surfaces are very common in Computer Aided Geometric Design and in this context it is important to obtain their implicit equations at a low computational cost. We will explain how to use syzygies to obtain implicit equations of parameterized surfaces. Implicitization algorithms via syzygies are more versatile and in many cases have faster running times than algorithms using Grobner bases and resultants. Our study will focus on the structure of the syzygies that determine the implicit equations for tensor product surfaces with basepoints.
Let d and k be positive integers and $\mu$ be a positive Borel measure on $\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. Using methods from convex optimization, we study the question of the minimal $m \in \mathbb{N}$ such that there exists a quadrature rule for $\mu$ with m nodes which is exact for all polynomials of degree at most $2d-1$ We show that if the support of $\mu$ is contained in an algebraic curve of degree $k$, then there exists a quadrature rule for $\mu$ with at most $dk$ many nodes all placed on the curve (and positive weights). This generalizes Gauss quadrature where the curve is a line and (the odd case of) Szegö quadrature where the curve is a circle to arbitrary plane algebraic curves. In the even case, i.e., $2d$ instead of $2d-1$ this result generalizes to compact curves. We use this result to show that, any plane measure $\mu$ has a quadrature rule with at most $3/2d(d-1)$ many nodes, which is exact up to degree $2d−1$.All our results are obtained by minimizing a certain linear functional on the polynomials of degree $2d$ and our proof uses both results from convex optimisation and from real algebraic geometry. (Joint work with Markus Schweighofer)
Counting (proper) colorings of graphs is a classical problem in combinatorics with connections to many different fields. In this talk, we want to examine proper $k$-colorings of graphs of the form $G \times P_n$, where $G$ is any graph, and where $P_n$ is the path graph on $n$ nodes. There are two special cases, namely (1) where $k$ is fixed but not $n$, and (2) where $n$ is fixed but not $k$. In (1), the number of colorings can be determined using the transfer-matrix method, and in (2), the number of colorings can be counted using the chromatic polynomial/Ehrhart theory. We will use the symmetry of $G \times P_n$ to combine the two methods to get explicit formulas (depending on $k$ and $n$) counting the number of colorings and we will give a restricted version of the famous reciprocity theorem for the chromatic polynomial. Furthermore, we will describe the doubly asymptotic behavior of graphs $G \times C_n$, where $C_n$ is the cycle graph with $n$ nodes as both $n$ and $k$ go to infinity. This is joint work with Alexander Engström.
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates). We suggest a geometric discretization which leads to factorisable discrete nets with a novel discrete analog of the orthogonality property and to an integrable discretization of the Euler-Poisson-Darboux equation. The coordinate functions of discrete confocal quadrics are computed explicitly. We demonstrate that special discrete confocal conics lead to incircular nets.
This is a joint work with W. Schief, Yu. Suris and J. Techter
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Computing tropical varieties is an algorithmically challenging task, requiring sophisticated techniques from computer algebra and convex geometry. We describe a new approach for computing zero-dimensional tropical varieties based on Newton polygon methods and apply it to computations of tropical starting points as well as tropical links. We show that, due to its nature, it is possible to obtain rigorous results in tropical geometry using finite precision in the algebraic computations.
A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares of quadratic forms. We show more generally that every nonnegative quadratic form on a real projective variety X of minimal degree is a sum of dim(X) + 1 squares of linear forms. This provides a new proof for one direction of a result due to Blekherman, Smith, and Velasco. We explain the geometry behind this generalization and discuss some related questions and open problems. (Joint work with G. Blekherman, R. Sinn, and C. Vinzant)
Linear Matrix Inequalities (LMI) are a class of convex feasibility problems appearing in different applied contexts. For instance, checking the asymptotic stability for linear differential systems, or computing nonnegativity certificates for multivariate polynomials, are LMI. I will discuss an approach based on techniques from real algebraic geometry to compute exact solutions to LMI, and what information are carried by this representation. The related algorithms are implemented in a Maple library called SPECTRA, and part of the talk will be dedicated to discuss tests on interesting examples.
An empty simplex is a lattice simplex with no lattice point except for its vertices. In my talk, based on a classification result by Batyrev and Hofscheier (2010) and with being inspired by the study on dilated 3-polytopes by Santos and Ziegler (2013), the existence of unimodular triangulations together with the integer decomposition property for odd dimensional dilated empty simplices will be discussed. No special knowledge is required to understand my talk.
Verbitsky's Global Torelli theorem has been one of the most important advances in the theory of holomorphic symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a version of the Global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly rich. We focus on birational contractions of symplectic varieties and generalize a number of known results for moduli spaces of sheaves to general deformations.
Our results are obtained through the interplay of Hodge theory, deformation theory, and a further instance of the concept "how to deduce beautiful consequences from ugly behavior of moduli spaces".
We consider the discrete logarithm problem of non-hyperelliptic curves. Observing that the computation can be sped up with the computation of special linear systems, we turn to the efficient computation of these. We then explore several techniques to do so. Finally, we come back to the discrete logarithm problem and present some theorems for curves of a fixed genus.
A beautiful theorem of J. Thas asserts that the ten points of a Desargues configuration of lines and points formed by two perspective triangles in the projective plane can serve as the ten nodes of a unique rational plane curve of degree 6. I will give an algebraic-geometric proof of this theorem by using the geometry of a special Coble surfaces obtained as blow-ups of ten nodes of a rational sextic. We will also discuss an interesting dynamics of automorphisms of these special Coble surfaces.
The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. Joint with Anna Seigal.
This talk deals with the interplay between tropical varieties and non-archimedean analytic varieties in the sense of Berkovich. The Berkovich space associated to a variety over a field with a non-archimedean valuation contains a lot of points which are not visible in algebraic geometry. Under certain conditions these additional points are helpful to find tropical varieties inside Berkovich spaces. After an introduction explaining tropicalizations and Berkovich spaces, we will discuss some results from joint work with Joseph Rabinoff and Walter Gubler.
In this talk I will discuss how computational algebraic geometry can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide two concrete examples of biological signalling systems that are better understood with the developed methods.
We consider the problem of determining the real rank of a real homogeneous polynomial f of degree d, i.e. the smallest possible number r such that f can be written as a sum of r dth powers of real linear forms. In geometric complexity theory the real rank of a polynomial is considered — among other notions of rank — as a measure of the computational complexity of evaluating the polynomial. After some general preliminaries we will present upper and lower bounds on the real rank of a monomial and we characterize those monomials for which the lower bound is attained. This is joint work with Enrico Carlini, Alessandro Oneto and Emanuele Ventura.
Model reduction of bio-chemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a bio-chemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors. Moreover, the the computed solution polytopes can be related to metastable regimes allowing to describe the qualitative dynamics of chemical reactions networks.
In this talk I present a complete solution to the dimension question for restricted Boltzmann machines (RBMs) that was first conjectured by Cueto, Morton, and Sturmfels in 2010. The RBM always has the expected dimension. This result comes from recent work with Jason Morton on the dimension and other properties of a much larger class of models called Kronecker product models: Take two sufficient statistics matrices A,B (in the RBM case, these are independence models, so Segre matrices). Take the Kronecker product of these matrices to obtain a new matrix F=A⊗BWhy care about the standard conjectures? (representing a new sufficient statistic), then marginalize the hidden variables corresponding to one of the original matrices. As this construction suggests, one can allow arbitrary exponential families (toric varieties) in place of the two independence models (Segre varieties) from the RBM.
Given a finite morphism between smooth projective curves one can associate to it a Prym variety (an abelian variety, not necessarily principal). The corresponding Prym map is the map between the moduli space of coverings and the moduli space of abelian varieties with some fixed polarization type. By dimension reasons, only in very few cases one can expect the Prym map to be generically finite over its image.
In this talk I will recall some notorious examples where the Prym map is finite and the geometrical interpretation of the fiber. I also explain the case of étale cyclic coverings of degree 7 over a genus 2-curve. We show that the Prym map is generically finite over a special subvariety of the moduli space of 6-dimensional abelian varieties with polarization type (1,1,1,1,1,7). By extending the map to a proper map on a partial compactification of the space of coverings and performing a local analysis we compute that the degree of this Prym map is 10.
This a joint work with Herbert Lange.
Hermann Schubert developed in the 19th century a calculus for answering enumerative questions in algebraic geometry, e.g., ''How many lines intersect four curves of degrees d_1,...,d_4 in three-dimensional space in general position?''. In his 15th problem, Hilberts asked for a rigorous foundation of Schubert's enumerative calculus, which led to important progress in algebraic geometry and topology (intersection theory of the Grassmannians). However, Schubert calculus only yields the typical number of complex solutions. Is there a meaningful way to speak about the typical number of REAL solutions?
We shall outline a way to do so, by assuming that the given objects (the four curves in the above example) are randomly rotated and to inquire about the expected number of real solutions. The approach blends ideas from real algebraic geometry with integral geometry and the theory of random polytopes.
(Joint work with Antonio Lerario.)
How can a discrete space be possibly curved? Because it shares some properties with Riemannian manifolds with particular curvature structures. In fact, in Riemannian geometry, sectional and Ricci curvature are central notions. They are defined infinitesimally, through first and second derivatives of the metric tensor. Curvature inequalities, like Sec 0, have geometric consequences, for the angles in geodesic triangles, convexity properties of the distance function, the divergence or convergence of geodesics, the growth of the volume of geodesic balls, the eigenvalues of the Laplace-Beltrami operator, or coupling properties of Brownian motion. In fact, it turns out that some of those local properties are equivalent to certain curvature bounds. Since such properties are also meaningful on metric spaces more general than Riemannian manifolds, we can use them to define corresponding curvature inequalities on such more general spaces, and to explore their consequences and mutual relations.
In this talk, I shall introduce or discuss some such curvature inequalities on suitable classes of metric spaces, in particular discrete ones, and explore their geometric consequences.
The study of cuts in graphs is an interesting topic in discrete mathematics with relations and applications to many other fields, such as algebraic geometry, combinatorial optimization or algebraic statistics. Here we focus on so-called cut polytopes and associated algebraic objects. We present some new results and open questions especially on algebraic properties of cut algebras and cut ideals.
The talk is based on joint work with Sara Saeedi Madani.
A combinatorial theory of linear systems on graphs and metric graphs has been introduced in analogy with the one on algebraic curves. The interplay is given by the Specialization Lemma. Let \(X\) be a smooth curve over the field of fractions of a complete discrete valuation ring and let \( \mathfrak{X} \) be a strongly semistable regular model of \(X\). It is possible to specialize a divisor on the curve to a divisor on the dual graph of the special fiber of \( \mathfrak{X} \); through this process the rank of the divisor can only increase. The complete graph \(K_d\) pops up if we take a model of a smooth plane curve of degree \(d\) degenerating to a union of \(d\) lines. Moreover omitting edges from \(K_d\) can be interpreted as resolving singularities of a plane curve. In this talk we present some results on linear systems on complete graphs and complete graphs with a small number of omitted edges, and we will compare them with the corresponding results on plane curves. In particular, we compute the gonality sequence of complete graphs and the gonality of graphs obtained by omitting edges. We explain how to lift these graphs to curves with the same gonality using models of plane curves with nodes and harmonic morphisms. This is partially a joint work with Filip Cools.
We present a large class of groups (no group known to be not in the class) that satisfy the Kervaire-Laudenbach Conjecture about solvability of non-singular equations over groups. We also show that certain singular equations with coefficients over groups in this class are always solvable. Our method is inspired by seminal work of Gerstenhaber-Rothaus, which was the key to prove the Kervaire-Laudenbach Conjecture for residually finite groups. Exploring the structure of the p-local homotopy type of the projective unitary group, we manage to show that many singular equations with coefficients in unitary groups can be solved in the unitary group. Joint work with Anton Klyachko.
Abelian varieties are group varieties, that is, loci given by polynomial equations which simultaneously admit a group structure. They are ubiquitous objects in algebraic and arithmetic geometry, as well as in number theory.
It is classically known that general abelian varieties of dimension at most five are Prym varieties associated to covers between algebraic curves. This reduces the study of abelian varieties of small dimension to the beautifully concrete and rich theory of curves.
I will discuss decisive recent progress on finding a structure theorem for abelian varieties of dimension six, and the implications this uniformization result has on the geometry of their moduli space.
How to ensure an outcome based on collective decisions is "better for all" when a) everyone acts in their personal interest only, and b) we do not know the exact "motives" of each person? Crudely, this is the central problem in mechanism design, a branch of economics. In this talk, I show how insights from tropical geometry can solve such problems.
The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format d×⋯×d, the complete semialgebraic description of the completable region is found.
We discuss two current projects at the interface of computer vision and algebraic geometry. Work with Joe Kileel, Zuzana Kukelov and Tomas Pajdla introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Work with Jean Ponce and Matthew Trager develops multi-view geometry for algebraic cameras that are represented by congruences in the Grassmannian of lines in 3-space.
Togliatti systems join in a beutiful way two seemingly unrelated topics: 1) weak Lefschetz property (WLP) and2) varieties with degenerate osculating spaces. The first examples were presented already in 1929 by Togliatti who studied monomial maps from P2 to P5. However, only recently Mezzetti, Miró-Roig and Ottaviani, using apolarity, proved general results relating projections of Veronese embedding with degenerate general osculating spaces and Artinian ideals that fail the WLP. In our talk we will present the above mentioned algebraic and geometric properties, providing further examples. We will define Togliatti systems and show new results, obtained jointly with Miró-Roig, on their classification, answering a conjecture of Ilardi (corrected by Mezzetti, Miró-Roig and Ottaviani).
