Castelnuovo's lemma states that d+3 points in general linear position in d-dimensional projective space uniquely determine a rational normal curve. However, the problem becomes more intricate for d+4 points, as it involves configurations of points in special position. The first notable examples of this phenomenon are Pascal’s and Brianchon’s theorems, which describe specific arrangements of six points on a conic in the projective plane. Pascal’s theorem asserts that six points A,...,F lie on a conic if the intersection points of the lines AB and DE, AF and CD, and EF and BC are collinear. Brianchon's theorem asserts that if six tangent lines to a conic are divided into two groups of three, the six vertices of the triangles formed by these groups also lie on a conic.
In this talk, we will present algebraic and combinatorial proofs of these classic results and discuss how these approaches can be extended to generalize Pascal’s and Brianchon’s theorems. Specifically, we will explore analogous statements for d+4 points on a rational normal curve of degree d. The results discussed are based on joint works with E. Carlini, N. Giansiracusa, H. Moon, and L. Schaffler.
The central objects in this talk are the descent polynomials of colored permutations on multisets, referred to as colored multiset Eulerian polynomials. These polynomials generalize the colored Eulerian polynomials that appear frequently in algebraic combinatorics and are known to admit desirable distributional properties, including real-rootedness, log-concavity, unimodality and the alternatingly increasing property. In joint work with Bin Han and Liam Solus, symmetric colored multiset Eulerian polynomials are identified and used to prove sufficient conditions for a colored multiset Eulerian polynomial to satisfy the self-interlacing property. This property implies that the polynomial obtains all of the aforementioned distributional properties as well as others, including bi-gamma-positivity. To derive these results, multivariate generalizations of a generating function identity due to MacMahon are deduced. The results are applied to a pair of questions, both previously studied in several special cases, that are seen to admit more general answers when framed in the context of colored multiset Eulerian polynomials. The first question pertains to s-Eulerian polynomials, and the second to interpretations of gamma-coefficients. We will see some of these results in detail, depending on the pace of the talk. At the last part of the talk, we will see connections between multiset permutations and polytopes from algebraic statistics.
Mirror symmetry gives a correspondence between Fano varieties and certain Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. To understand the classification using this correspondence, we ask the question: ‘Which pairs of Laurent polynomials (f,g) are mirrors to pairs of Fano varieties that are related by a blow-up X_g --> X_f?’ The combinatorial criteria on (f,g) we are looking for will generalise the relationship between the fans of two toric varieties related by a toric blow-up.
I will present a new method of constructing Fano mirrors to Laurent polynomials, which uses mirror constructions from the Gross-Siebert program. This will include the construction of a birational morphism between the mirrors to a pair of Laurent polynomials in two variables that satisfies certain combinatorial conditions.
We present a general approach for the study of dimer model limit shape problems via variational and integrable systems techniques. In particular we deduce the limit shape of the Aztec diamond and the hexagon for quasi-periodic weights through purely variational techniques.
Putting an M-curve at the center of the construction allows one to define weights and algebro-geometric structures describing the behavior of the corresponding dimer model. We extend the quasi-periodic setup of our previous paper [7] to include a diffeomorphism from the spectral data to the liquid region of the dimer.
Our novel method of proof is purely variational and exploits a duality between the dimer height function and its dual magnetic tension minimizer and applies to dimers with gas regions. We apply this to the Aztec diamond and hexagon domains to obtain explicit expressions for the complex structure of the liquid region of the dimer as well as the height function and its dual.
We compute the weights and the limit shapes numerically using the Schottky uniformization technique. Simulations and predicted results match completely.
In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called “Wasserstein polytopes” or “Kantorovich-Rubinstein polytopes” in the literature. Recently such polytopes have been shown to play an important role in a host of different contexts – however, little is known to date about their structure.
In particular, Vershik asked about the stratification of the metric cone according to the combinatorial type of such polytopes.
After stating the definitions and some examples, in this talk I will define an arrangement of hyperplanes that describes the stratification sought by Vershik, together with some computational results on enumerative invariants in the case of metrics on up to six points.
This will also allow us to compare Wasserstein polytopes with "Tight spans”, showing that the stratifications of the metric cone induced by these two combinatorial invariants are not related by refinement. Time permitting, I will mention some open problems.
The talk is based on joint work with Lukas Kühne and Leonie Mühlherr.
Generalized weights are a well-studied family of invariants, originally defined for linear block codes in 1977. Their importance stems from the fact that they characterize the code robustness against wiretapping. After a short introduction to the topic, in this talk we will investigate which sequences of integers can be the generalized weights of various types of error-correcting codes: linear block codes, rank-metric codes, and the more general case of sum-rank metric codes.
In this talk, we study tensors that can be decomposed via successive rank-one approximations. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. We study the tensors that can be decomposed by the following procedure: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the rank. Moreover, this decomposition may depend on the order in which terms are subtracted. However, if all singular vectors in the process are orthogonal in at least two factors, then they are all singular vector tuples of the original tensor, and the decomposition is valid independent of order. Tensors that admit such a decomposition are called two-orthogonal and we study the geometry of the variety of these tensors.
When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations.
Joint work with Heather Harrington (Oxford/MPI Dresden) and Mike Stillman (Cornell)
Tropicalization is a process that associates to an algebro-geometric object a piecewise linear polyhedral shadow that captures its essential combinatorial structure. In this talk, I will give an overview of the numerous ways on how to extract tropical information from a matrix over a non-Archimedean field. Each perspective will give rise to inherently quite different phenomena. Central instances of this rich panorama include the tropical geometry of vector bundles, logarithmic concavity results for valuated (bi-)matroids (using techniques from combinatorial Hodge theory), and the geometry of affine buildings.
This talk draws from joint work with Andreas Gross and Dmitry Zakharov; Andreas Gross, Inder Kaur, and Annette Werner; Felix Röhrle; Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis; Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond Coles.
Signomials generalize polynomials by allowing arbitrary real exponents, at the expense of restricting the resulting function to the positive orthant. In this talk, I present a signomial Positivstellensatz based on conditional "sums of arithmetic-geometric exponentials" (SAGE). The Positivstellensatz applies to compact sets which need not be convex or even basic semi-algebraic. In the first part of the talk, I explain how this result is derived through the newly-defined concept of signomial rings. Then I show how the same concept leads to a novel convex relaxation hierarchy of lower bounds for signomial optimization. These relaxations (which are based on relative entropy programming) can be solved more reliably than those arising from earlier SAGE-based Positivstellensätze. Moreover, this increase in reliability comes at no apparent cost of longer solver runtimes or worse bounds. Numerical examples are provided to illustrate the performance of the hierarchy on problems in chemical engineering and reaction networks. To conclude, I provide an outlook on how any (hierarchical) inner-approximation of the signomial nonnegativity cone yields upper bounds for signomial optimization.
This talk is based on joint work with Riley Murray.
In 1999 Bayer-Popescu-Sturmfels constructed minimal free resolutions of binomial ideals which correspond to the diagonal embeddings of unimodular toric varieties. Such resolutions are significant in algorithms concerning homological algebra and algebraic geometry. In this talk I will tell the story of how researchers from commutative algebra and homological mirror symmetry joined forces to generalize this construction to all smooth projective toric varieties.
Hauenstein and collaborators recently introduced a numerical algorithm to identify the connected components in an arrangement of real hypersurfaces. We explain the main ideas, which are based on Morse theory, and we present novel refinements, using the likelihood equations. We offer an implementation in Julia, and we discuss an ongoing case study for arrangements of Schubert divisors in the Grassmannian of lines in 3-space.
Let D be a symmetric domain and G a discrete algebraic group acting on it. For example, one can take the 2-by-2 integral special linear group acting on the Siegel upper half-plane via Mobius transformations. We consider a certain (toroidal) compactification of the quotient space D/G, due to Mumford, and we describe its tropicalization, as follows. We construct a collection of polyhedral cones that are in dimension-reversing bijection with the boundary components of our compactification. You can think of these polyhedral cones as encoding the combinatorics of the boundary of our compactification. Finally, I will show how this tropical object can be used to deduce statements about the cohomology of the original space. The main example I will discuss will be the moduli space of tropical abelian varieties.
Based on joint work in progress with E. Assaf, M. Brandt, J. Bruce, and M. Chan.
Every partially symmetric tensor can be expressed as a linear combination of a finite number of so-called decomposable partially symmetric tensors. The rank of a partially symmetric tensor is defined as the smallest positive integer r such that the partially symmetric tensor can be written as a linear combination of r decomposable partially symmetric tensors. In this talk, we discuss an algebro-geometric approach to the problem of finding the generic rank of partially symmetric tensors, that is, the rank of a generic partially symmetric tensor.
Motivated by the great riddle in game theory of establishing cooperation as a rational solution in the Prisoner’s Dilemma, the philosopher Wolfgang Spohn poses a new equilibrium concept: dependency equilibria.
In this talk we study dependency equilibria from an algebraic viewpoint as the probabilities inside the so-called Spohn variety and, for 2x2 games, explore the possible density of real points within this variety by looking at its planar model.
Network games are an important class of games that model agent interactions in networked systems, where players are situated at the nodes of a graph and their payoffs depend on the actions taken by their neighbors. We extend the classical framework by considering a game model where the strategies are positive semidefinite matrices having trace one. These (continuous) games can serve as a simple model of quantum strategic interactions. We focus on the zero-sum case, where the sum of all players’ payoffs is equal to zero. We establish that in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron, that is, the feasible region of a semidefinite program. Furthermore, we demonstrate that determining whether a game is a semidefinite network game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.
On the right side of the blue MEGA poster, we see three plaster models that were built around 1890 by PhD students of Sophus Lie at Leipzig University. This thread was picked up in the 2021 paper on Theta Surfaces with Daniele Agostini, Türkü Celik and Julia Struwe. This talk is an introduction that merges historical and current perspectives.
A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a fixed phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML degree of the BMT model on a star tree with $n+1$ leaves is $2^n+1 - 2n - 3$, which was previously conjectured by Amendola and Zwiernik. This talk will focus on a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Prufer theorem to complete graphs with weights given by a tree, and on the intersection theory used to compute the ML-degree of the star tree.
A fundamental computational question in physics is to understand the equilibrium state of a system composed of many particles. In the setting of quantum mechanics, the system is described by a self-adjoint operator, so-called Hamiltonian, and the goal is to compute its minimum eigenvalue or ground energy. After describing the problem in detail, I will explain how tools from convex optimization such as semidefinite programming and sums of squares relaxation can be used to study it, and if time permits, how new entropy constraints can be used to strengthen such relaxations.
We propose an effective algorithm to decide if a homogeneous prime ideal can be made toric by a linear automorphism of the ambient space. If this is the case, the algorithm computes such a transformation explicitly. We benchmark the algorithm on Gaussian graphical models on five vertices and Gaussian conditional independence models of undirected graphs up to six vertices. We find that these are either toric from the start or cannot be made toric by linear coordinate changes. This is joint work with Julian Vill.
I will discuss the fundamental problem of degree sequence optimization and its special case of matrix line sum optimization, which are wide open. I will suggest some of the many open problems, describe some partial results such as for convex polynomials, and show some of the techniques used, including our recent powerful theory of sparse integer programming in high dimensions, which will be explained.
In the last fifty years quiver representations have been increasingly used in various fields of mathematics, from representation theory to topological data analysis, to algebraic geometry.
In this talk, we will give an introduction to the theory of their moduli spaces and to several topics of active research within it.
We will construct these moduli spaces using King's geometric invariant theory and overview their relation with the representation theoretical properties of quivers.
Several parallels with the geometry of moduli of vector bundles on algebraic curves will be drawn, both showcasing analogous results and similar steps in their proofs.
Time permitting, we will describe several aspects of the geometry of these quiver moduli, such as their deformation theory, their birational properties, and the description of their derived categories.
This work is supported by the Luxembourg National Research Fund (FNR-17953441).
In the 19th century, Steiner asked: How many smooth conics are tangent to five given conics in the plane? His original answer, $6^5$, turned out to be wrong. The issue of correctly computing numbers like this and rigorously proving them fueled much of the development of modern intersection theory and enumerative geometry. We will see where the problem with Steiner's answer lies and how to solve similar problems that were open until recently, for instance: How many smooth cubic surfaces are tangent to 19 lines in projective 3-space? I will also explain the general notions of characteristic numbers and excess intersection. Time permitting, we might even outline some other typical approaches, like degenerations and compactifications admitting a modular interpretation. This is partly joint work with Mara Belotti, Alessandro Danelon, Jiahe Deng and Claudia Fevola.
The high dimensional eigenvalue problem that encodes the electronic Schrödinger equation can be approximated by a hierarchy of polynomial systems at various levels of truncation, called the coupled cluster (CC) equations. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We determine the number of complex solutions to the CC equations over the Grassmannian. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.
We introduce the class of matroidal polynomials. They are (possibly inhomogeneous) polynomials attached to a matroid and axiomatized by Deletion-Contraction identities. When homogeneous they are matroid support polynomials--polynomials whose monomial support are the bases of a matroid. We show that matroidal polynomials have rational singularities, provided the underlying matroid is connected of rank at least two. The proof is jet-theoretic and involves controlling the dimension of certain jet loci.
By similar methods we show flag matroidal polynomials (attached to a flag of matroids) and Feynman integrands (attached to a Feynman diagram) have rational singularities. When the Feynman diagram satisfies general kinematics, this proves: the Feynman integral in Lee-Pomeransky form is the Mellin transformation of a polynomial with rational singularities.
Our goal is to explain most of these concepts as well as the matrimony between matroids and jets.
Joint with Uli Walther.
In the area of causal discovery, one seeks to use data to learn the directed acyclic graph (DAG) that best explains the causal relationship among random variables. One method of learning this graph is via a linear program over the characteristic imset (CIM) polytope. We introduce an interventional analogue of the CIM polytope and show that interventional data can be used to learn the DAG via a linear program. We also use this framework to compute an H-representation for faces of CIM polytopes given by restricting graphs to have a fixed tree skeleton.
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 Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This dual volume replaces other natural objective functions in convex optimization, such as the logarithmic barrier minimized by the analytic center. When translating the facet hyperplanes, the Santaló point traces out a patchwork of semialgebraic sets. I will describe and compute this geometry using algebraic and numerical techniques. I will also explore connections with statistics, optimization and physics. This is joint work with Dmitrii Pavlov.
One of the most fundamental questions in tropical geometry is: How much topological information does the tropicalization of a variety retain.
Although Viro patchworking was one of the earliest achievements in tropical geometry, the area of real tropical geometry remains comparatively unexplored. In this talk we study real tropicalizations of discriminants, complete intersections, and steady state varieties from reaction networks.
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 present an ongoing work in progress with Simon Telen where we study amplituhedra contained in the Grassmannian Gr(2,4) of lines in projective 3-space. We are particularly interested in geometric descriptions, the algebraic boundary, and the adjoint which should correspond to the numerator of the canonical form in the sense of positive geometry.
The amplituhedron is an object defined by physicists to understand particle scattering, and has garnered much recent attention from physicists and mathematicians alike. Mathematically, it is a linear projection of a positive Grassmannian to a smaller Grassmannian, via a totally positive matrix. Generalizing the amplituhedron, we define a Grasstope to be such a projection by any matrix.
In this talk we ask: can we gain new insights by broadening our horizons, and systematically studying all Grasstopes? I will explain some current work with Yelena Mandelstam and Dmitrii Pavlov in this direction.
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.
The spectrum of the Jacobian matrix G of a dynamical system plays a central role in the stability and bifurcation analysis of equilibria. In particular, a complex pair of purely imaginary eigenvalues of G is a necessary condition for Hopf bifurcation and consequent oscillatory behavior.
Reaction networks give rise to parametric systems of equations, and thus to parametric families of Jacobian matrices. In this talk I share our work-in-progress about characterizing the networks for which there is no choice of parameters such that G possesses purely imaginary eigenvalues.
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.
