The concept of multidegrees provides the right generalization of degree to a multiprojective setting, and its study goes back to seminal work by van der Waerden in 1929. We will give a basic introduction to the notion of multidegrees of a multiprojective variety. Then, a complete characterization for the positivity of multidegrees will be presented. Finally, we will review some important applications that derive from the use of multidegrees.
We show how numerical methods from nonlinear algebra can be used to study scattering amplitudes in particle physics. The story involves solving systems of polynomial equations, computing maximum likelihood degrees of point configuration spaces, and numerical elimination. We discuss problems that were solved using these techniques and were previously out of reach.
The notion of degenerations has been at the forefront of algebraic geometry for the past decades. A particularly interesting class are so-called Mustafin degenerations, which are degenerations of projective varieties induced by combinatorial data in a Bruhat–Tits building. They were introduced in the 1970s by Mustafin in the context of uniformisation and have since found applications in several areas of algebraic and arithmetic geometry. In this talk, we give a classification of Mustafin degenerations in terms of combinatorial data and outline applications to p-adic non-abelian Hodge theory, computer vision and coding theory.
Many exciting research topics lie at the interface between polyhedral and algebraic geometry creating fruitful grounds for new computational methods. Tropical geometry has recently fueled these interactions. At the same time, computer-oriented approaches raise novel questions in the analysis of algebraic and geometric objects. In this talk, I exhibit examples of these aspects based on my research. I will describe polyhedral computations at the core of the study of curves, surfaces and triangulations through tropical lenses, and algebraic properties of constrained realization spaces of polytopes.
There is a series of four long-standing conjectures on group rings that are attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group over a field has no zero divisors. I will discuss my recent disproof of the unit conjecture and the connections between the remaining three Kaplansky conjectures and open questions about $L^2$-invariants, K-theory and geometric group theory.
Tensor networks arise in the quantum many-body physics to study systems with a particular combinatorial structure. Similar mathematical objects are central in several other areas of pure and applied mathematics.
In this talk, I will provide an overview of some recent achievements obtained using geometric methods. I will briefly focus on two aspects: characterizing membership in certain varieties of interest via representation theory, and using variational techniques and classical algebraic geometry to obtain elements satisfying a desired optimality condition.
Toric varieties are beautiful objects in Algebraic Geometry whose structure can be studied via their connection to lattice polytopes. They are also varieties that admit a simple parameterization via monomials. Their simple, yet rich structure means that they are versatile objects that appear in many applied settings. In statistics, log-linear models are toric varieties. In Geometric Modeling, the connection to polytopes is used to define bases of polynomials whose domains are lattice polytopes and which have useful approximation properties.
In this talk we explain how understanding properties of toric varieties gives insight into their practical use in Statistics and Geometric Modeling and how the toric geometry language in fact brings these two areas together.
Systems of linear differential equations and their solutions are ubiquitous in applications, and so are their algebraic counterparts: ideals of linear differential operators and holonomic functions. In this talk, I explain how this theory enhances the computational toolbox as well as the structural understanding of several problems in the sciences ranging from the statistical inference of data to scattering processes in particle physics.
Measure valued solutions to constant coefficient PDEs appear natural in calculus of variations. For instance they can be used characterise gradients, symmetric gradients or appear in the Euler-Lagrange equations of geometric variational problems. In this talk I discuss how the wave cone -- the real variety determined by the kernel of the associated polynomial matrices -- gives structural conditions on the “size” and “directions” of the singular part of the measure. For instance we can conclude in certain situations the rectifiability of the “lowest density” part of the measure.
Reaction networks with mass-action kinetics give rise to high-dimensional polynomial ODE systems with positive parameters. Chemical reaction network theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions depending on the underlying network structure alone. However, mass-action kinetics holds only for elementary reactions in homogeneous and dilute solutions. In joint work with Stefan Müller, we propose the notion of generalized mass-action systems, which can serve as a more realistic model for biochemical reaction networks. It turns out that in this setting, uniqueness and existence of steady states for all parameters additionally depend on sign vectors of related real subspaces. In terms of the corresponding generalized polynomial equations, our results guarantee existence and uniqueness of positive solutions for all positive parameters.
In this talk, we describe some of our results for positive solutions of generalized polynomial equations and the motivation coming from the study of reaction networks. We also outline recent joint work on when the existence of a unique positive solution is robust with respect to small perturbations of the real exponents and on the stability of the resulting steady state. Finally, we discuss directions of ongoing and future research for positive solutions of generalized polynomial equations and applications to reaction networks.
It was conjectured by White in 1980 that the toric ring associated to amatroid is defined by symmetric exchange relations. These are degree two relations reflecting the combinatorial structure of a matroid. This conjecture was extended to discrete polymatroids by Herzog and Hibi. Several special cases of the conjecture has been proved, but it remains open in full generality. In this talk I will give an introduction to discrete polymatroids, White's conjecture, and related problems.
A fundamental task in modern artificial intelligence is to identify transparent ways to represent cause-effect relations and then design efficient and reliable methods for learning such representations from data. These tasks can, respectively, be termed the problem of representation and the problem of causal discovery, and each problem has close connections to the world of nonlinear algebra. Classically, the problem of representation is solved using directed acyclic graphical (DAG) models, which are then learned from data using a variety of techniques - among which the most popular is greedy search. After taking a geometric view of this classical story, we will dive into the newest trends in causal discovery algorithms, which rely on a combination of observational and interventional data to learn a causal DAG. To understand the geometry and algebra of such causal models, we will broaden our perspective beyond DAGs to the family of staged trees. This new perspective will not only allow us to generalize and unify some previous results on the algebraic geometry of DAG models and staged trees, but it will also motivate a new family of context-specific causal models, called CStrees, that admit nice representation theorems analogous to those of DAGs. Time permitting, we will discuss the statistical theory of these new models, and see some applications to real data.
Descartes rule of signs for univariate real polynomials is a beautifully simple upper bound for the number of positive real roots. Moreover, it gives the exact number of positive real roots when the polynomial is real rooted, for instance, for characteristic polynomials of symmetric matrices. A general multivariate Descartes rule is certainly more complex and still elusive. I will recall the few known multivariate cases and will present a new optimal Descartes rule for polynomials supported on circuits, obtained in collaboration with Frédéric Bihan and Jens Forsgård.
An ODE model with parameters is said to be structurally identifiable if the values of parameter can be uniquely determined from continuous noise-free data. This property is a natural prerequisite for practical identifiability. It may happen that, although the model is not identifiable, it becomes identifiable if several independent experiments (with the same parameter values but different initial conditions) are conducted. A natural question is: how many experiments are sufficient to get "the maximal possible identifiability"?
We give an algorithm for computing a bound for the number of experiments "providing the maximal possible identifiability" which is off at most by one. The algorithm is fast in practice (and has polynomial arithmetic complexity). The algorithm is based on our theoretical results about expressing the field of definition of a (differential-algebraic) variety in terms of several its generic points. Interestingly, the process of discovery and establishing of these properties originated from model theory.
This is joint work with Alexey Ovchinnikov, Anand Pillay, and Thomas Scanlon.
A homogeneous polynomial F admits an SOS-multiplier certificate on a real projective variety X if there exist sums of squares g and s such that Fg=s in the homogeneous coordinate ring of X. Such an expression certifies the nonnegativity of F and is thus of considerable theoretical and practical importance. In this talk I will present ongoing work with G. Blekherman, R. Sinn and G.G. Smith on the geometry of such nonnegativity certificates. Our main results are effective bounds on the degrees of possible g's on algebraic curves and surfaces which depend on classical geometric invariants of the varieties. These bounds are, in some cases, provably optimal.
Given a subspace arrangement we may associate to it two ideals, the intersection of the linear ideals associated to each subspace or their product. The structure of the intersection ideal is mostly unknown. For example already for a finite collection of generic points in a projective space the degree of the generators of the intersection is not known. On the other hand the product ideal is better understood. An old theorem of Herzog and myself asserts that the product ideal has a linear resolution, or, which is the same, its regularity is given by the number of factors. In the talk we will discuss the structure of the resolution of the product ideal. We will see that such a resolution is supported on a polymatroid.
In collaboration with Manolis Tsakiris of ShanghaiTech.
Chemical reaction networks are often used to model and understand biological processes such as cell signaling. Under the assumption of mass action kinetics, reaction networks give rise to polynomial dynamical systems. The ideals generated by these polynomials are called steady-state ideals. Steady-state ideals appear in multiple contexts within the chemical reaction network literature, however they have yet to be systematically studied. To begin such a study, we ask and partially answer the following question: when do two reaction networks give rise to the same steady-state ideal? In this talk, we will describe three operations on the reaction networks that preserve the steady-state ideal. Furthermore, we will describe combinatorial conditions to identify monomials in a steady-state ideal.
A Segre product of projective spaces X describes all decomposable tensors with a prescribed format. When the dual variety of X is a hypersurface, its defining polynomial is the classical hyperdeterminant. In the first part of this talk, we discuss the asymptotic behavior of the degree of the hyperdeterminant for tensors of hypercubical format.
Another fundamental invariant attached to the Segre product X is its Euclidean Distance Degree (ED degree), which measures the complexity of minimizing the distance from X. Friedland and Ottaviani derived a beautiful formula for the ED degree of X and observed that this invariant stabilizes as soon as the dimension of one of the factors is large enough. In the second part of this talk, we give a more geometric explanation of this fact.
In the last part, we consider the Segre product Z between a projective variety Y and a smooth quadric hypersurface Q. We discuss the stabilization of the degree of the dual variety of Z as soon as the dimension of the quadric is large enough. This is joint work with Giorgio Ottaviani and Emanuele Ventura.
I will discuss recent work calculating the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of $A_g$ and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
Maximum likelihood estimation is an optimization problem used to fit empirical data to a statistical model. The number of complex critical points to this problem when using generic data is the maximum likelihood degree (ML-degree) of the model. The concentration matrices of certain models form a spectrahedron in the space of symmetric matrices, defined by the intersection of a linear subspace $\mathcal{L}$ with the cone of positive definite matrices. It is known what the ML-degree should be for such models when $\mathcal{L}$ is generic. In this talk I will describe the 'non-generic' linear subspaces, that is, those for which the corresponding model has ML-degree lower than expected. More specifically, for fixed $k$ and $n$, I will describe the geometry of the Zariski closure in the Grassmanian $G$ $(k,($$\substack{n+1\\2}$$))$ of the set of $k$-dimensional linear subspaces of symmetric $n$ $\times$ $n$ matrices that are 'non-generic' in this sense. I will show that this closed set coincides with the set of linear subspaces of symmetric matrices for which strong duality in semi-definite programming fails. This is joint work with Yuhan Jiang and Kathlén Kohn.
The Pauli exclusion principle is fundamentally important for our understanding of matter since it restricts the way electrons at zero temperature can distribute in space. At first sight, extending the respective physical theories for systems of N electrons beyond their ground state regime seems to be straightforward. Yet, in the form of the so-called one-body N-representability problem a non-trivial compatibility problem emerges. In our presentation, we explain how concepts of convex geometry allow us to solve that long-standing problem, leading to a hierarchy of generalized exclusion principle constraints. For this, we illustrate in particular how standard concepts of quantum many-body physics translate into convex geometry as lineups, threshold complexes, Gale posets and permutation-invariant polytopes.
FI is the category of finite sets with injections. An FI something F is a functor from FI to the category of somethings. In particular, for S in FI, the symmetric group Sym(S) acts on F(S) by something automorphisms of F(S). FI-somethings arise in several areas of mathematics; for instance, FI-modules arise as cohomology groups of configuration spaces of n labelled points on a given manifold (with n varying), and FI-algebras arise as coordinate rings of certain algebraic-statistical models, where the number of observed random variables varies. This talk aims to be a gentle introduction into algebra over FI; no prior exposure to FI-maths is required.Sometimes, FI-somethings inherit good properties of the category of somethings (and indeed the same applies when FI is replaced by other suitable base categories). I will present several, by now, classical examples of this phenomenon, primarily due to Church, Ellenberg, Farb, Daniel Cohen, Aschenbrenner, Hillar, Sullivant, Sam, and Snowden. For instance, a finitely generated FI-module M over a Noetherian ring is Noetherian, and if the ring is a field, then the dimension of M(S) is eventually a polynomial in |S|.After this overview, I will zoom in on new joint work with Rob Eggermont and Azhar Farooq that says that if X(S) < K^{c x S} is avariety of c x S-matrices and for every injection S -> T the natural map K^{c x T} -> K^{c x S} maps X(T) into X(S), then the number of Sym(S) orbits on the set of irreducible components of X(S) is a *quasi*polynomial in |S| for |S| sufficiently large.
Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible testbed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Süß and Hausen, and Gonzales showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzales, and Süß showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective space, in particular the blow-ups of general arrangements of points studied by Castravet and Tevelev. In this talk I'll review some of these results, and then show a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data. I'll describe new examples and pose some questions. This is joint work with Kiumars Kaveh.
Mustafin varieties are degenerations of projective spaces induced by combinatorial data in a Bruhat--Tits building. In recent years, these objects have proved to be of great interest due to a wide array of applications in algebraic and arithmetic geometry. In this talk, we review these developments. Moreover, we introduce a generalisation - that we call Mustafin degeneration - which is obtained by considering combinatorial data in a compactification of the respective Bruhat-Tits building. This new notion provides a novel combinatorial framework for the study of one parameter families of so-called multiview varieties in computer vision.
This talk is partially based on joint work with Binglin Li and Annette Werner.
A continuous map $f:\mathbb{C}^n\longrightarrow \mathbb{C}^N$ is called $k$-regular, if the image of any $k$ distinct points in $\mathbb{C}^N$ are linearly independent. The study of the existence of regular maps was initiated by Borusk 1957, and later attracted attention due to its connection with the existence of interpolation spaces in approximation theory, and certain inverse vector bundles in algebraic topology.In this talk, based on a joint work with Joachim Jelisiejew, we consider the general problem of the existence of regular maps to Grassmannians $\mathbb{C}^n\longrightarrow Gr(\tau,\mathbb{C}^N)$. We will discuss the tools and methods of algebra and algebraic geometry to provide an upper bound on $N$, for which a regular map exists.
The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, by work of Herrmann-Jensen-Joswig-Sturmfels, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces -- the positive Dressian, and the positive tropical Grassmannian -- so it is natural to ask how these two positive spaces compare. I'll discuss joint work with David Speyer in which we showed that the positive Dressian equals the positive tropical Grassmannian. This result has several applications, for example a new "tropical" proof of da Silva's 1987 conjecture that all positively oriented matroids are realizable. In joint work with Lukowski and Parisi, we showed that the positive Dressian controls positroidal subdivisions of the hypersimplex, and furthermore showed that there is a striking parallel between positroidal subdivisions of the hypersimplex, and positroidal subdivisions of the m=2 amplituhedron.
Despite the title of the seminar, "Nonlinear Algebra Seminar", we focus in this talk on linear degenerations of the flag variety, which does appear prominently not only in representation theory. The flag vaeriety could be defined by the vanishing of Plücker relations, while by considering the flag variety in terms of Linear Algebra, the notion of quivers and their representations, and, following this, quiver grassmannians are naturally. A linear degeneration of the flag variety is then a degeneration of the flag variety when considered as a quiver Grassmannian of type A.
In the talk we discuss an order along the degenerations, the flat (irreducible) degenerations, meet the Catalan numbers and translate our degenerations into degenerate Plücker relations. This provides a link to the tropical flag variety *and maybe an approach how to compute the latter in some future*. (*means "dreaming")
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters are generically identifiable for 2-tree CFN and K3P mixtures. We also show that the k-cycle phylogenetic network parameter is identifiable under the K2P and K3P models. This is joint work with Benjamin Hollering.
In this talk I will investigate the structure of the "moduli space" $W(G,d)$ of a geometric graph $G$, i.e. the set of all possible geometric realizations in $R^d$ of a given graph $G$ on n vertices. Such moduli space is Spanier-Whitehead dual to a real algebraic discriminant. For instance, in the case of geometric realizations of $G$ on the real line, the moduli space $W(G, 1)$ is a component of the complement of a hyperplane arrangement in $R^n$. Numerous questions about graph enumeration can be formulated in terms of the topology of this moduli space.I will explain how to associate to a graph $G$ a new graph invariant which encodes the asymptotic structure of the moduli space when d goes to infinity. Surprisingly, the sum of the Betti numbers of $W(G, d)$ stabilizes, as d goes to infinity, and gives the claimed graph invariant $B(G)$ -- even though the cohomology of $W(G, d)$ "shifts" its dimension (we call the invariant $B(G)$ the "Floer number" of the graph $G$, as its construction is reminiscent of Floer theory from symplectic geometry.)This is joint work with M. Belotti and A. Newman
Algebraic models of data can be interpreted as a point-sample of algebraic varieties. Sampling varieties is a powerful tool to recover its topology and hence its shape. I will give a brief introduction to sampling and the key role played by the bottleneck degree of the variety.
Tropical varieties are polyhedral shadows of algebraic varieties which allow to study the latter with combinatorial methods. In this talk we will focus on tropicalizations of Grassmannians, and in particular the tropical Grassmannian $TGr_0(3,8)$. It might happen that the intersection of finitely many tropical hypersurfaces is not a tropical variety. Such a set is called a tropical prevariety. Examples of those prevarieties are Dressians, which contain tropical Grassmannians and are moduli space of tropical linear spaces. Tropical linear spaces, tropical Grassmannians and Dressians naturally appear in many mathematical areas, and also in computer science, biology, economics and physics. More precisely in the study of shortest paths, phylogenetics, auctions and scattering amplitudes. The later is directly connected to the non-negative part of the tropical Grassmannians. I will intoduce tropical Grassmannians and Dressians and discuss differences between different polyhedral structures on the former with a focus on the case $TGr_0(3,8)$.
The problem of recovering a row or column sparse low rank matrix from linear measurements arises for instance in sparse blind deconvolution. The ideal goal is to ensure recovery using only a minimal number of measurements with respect to the joint low-rank and sparsity constraint. We consider gradient based optimization methods for this task that combine ideas of hard and soft thresholding with Riemannian optimization. This is joint work with Henrik Eisenmann, Felix Krahmer and Max Pfeffer.
The polynomial ring in n variables is a direct sum of one-dimensional vector spaces spanned by monomials. Modules that are similarly multigraded as direct sums of finite dimensional vector spaces over the integer lattice enjoy concrete, effective algebra. For example, monomial ideals have unique minimal primary decompositions as intersections of primary monomial ideals. What is it about the integer lattice that drives this good fortune? Would the same work if, say, the underlying multigrading were a direct sum over a real vector space instead? The ambient ring would then consist of real-exponent polynomials, and the answer has important implications for applications to topological data analysis. This talk explores how far the hypotheses for multigradings can be relaxed while maintaining fundamental results from commutative and homological algebra, with a particular focus on primary decomposition.
A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points. Stable polynomials have close relations to matroids and hyperbolic programming. We will discuss a generalization of stability to algebraic varieties of codimension larger than one. They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.
Homotopy continuation is an important technique for solving systems of polynomial equations numerically. A basic approach is to track the Bézout number many paths in an affine space. More advanced strategies include polyhedral homotopies and (multi) projective homotopies. The former exploits the polyhedral structure of the equations to reduce the number of paths, while the latter avoids diverging paths by tracking in a compact space. We combine the advantages of these two strategies by tracking paths in a compact toric variety, naturally associated to our polynomial system. A quotient construction by Cox allows us to use global coordinates on this toric variety. I will explain the main ideas and illustrate the advantages of our approach by means of examples. This is joint work with Timothy Duff, Elise Walker and Thomas Yahl.
Tensor decomposition has many applications. However, it is often a hard problem. Orthogonally decomposable tensors form are a small subfamily of tensors and retain many of the nice properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. We will see that the decomposition of such tensors can be found efficiently, their eigenvectors can be computed efficiently, and the set of orthogonally decomposable tensors of low rank is closed and can be described by a set of quadratic equations. One of the issues with orthogonally decomposable tensors, however, is that they form a very small subset of the set of all tensors. We expand this subset and consider orthogonally decomposable tensor trains. These are formed by placing an orthogonally decomposable tensor at each of the vertices of a tensor train, and then contracting. We give algorithms for decomposing such tensors both in the setting where the tensors at the vertices of the tensor train are symmetric and non-symmetric. This is based on joint work with Karim Halaseh and Tommi Muller.
A tree amplituhedron is a geometric object generalizing the cyclic polytope and the positive Grassmannian. It was introduced by Arkani-Hamed and Trnka to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. In particular, the physical computation of scattering amplitudes is reduced to finding the triangulations of the amplituhedron. I will start with a gentle overview of the amplituhedron. Then, I will explain how to find its triangulations in various cases. As amplituhedron is in the heart of the general theory of positive geometry, I will present some related examples of positive geometries arising in physics, as well. This is joint work with Leonid Monin and Matteo Parisi.
The moduli space of smooth curves is one of the fundamental concepts in algebraic geometry. Severi conjectured that this space would be rational in every genus, meaning that we could write down a general Riemann surface in terms of parameters, such as the coefficients in the equation of a plane curve. This was spectacularly disproven by Harris and Mumford in the 80s and our knowledge on the birational geometry of this space has increased a lot since then. In my talk, I will present this circle of ideas, together with some recent results obtained together with Ignacio Barros.
Holonomic techniques have deep roots going back to Wallis, Euler, and Gauss, and have evolved in modern times as an important subfield of computer algebra, thanks in large part to the work of Zeilberger and others over the past three decades. In this talk, I will give an overview of the area, and in particular will present a select survey of known and original results on decision problems for holonomic sequences and functions. I will also discuss some surprising connections to the theory of periods and exponential periods, which are classical objects of study in algebraic geometry and number theory; in particular, I will relate the decidability of certain decision problems for holonomic sequences to deep conjectures about periods and exponential periods, notably those due to Kontsevich and Zagier.
Parts of this talk will be based on the paper "On Positivity and Minimality for Second-Order Holonomic Sequences".
Reaction networks (i.e., oriented graphs embedded in Euclidean space) give rise to systems of differential equations with polynomial right-hand side. In general, such systems are very difficult to analyze. For example, they can give rise to questions related to Hilbert’s 16th problem, the Jacobian conjecture, or chaotic dynamics.
We will discuss algebraic approaches for analyzing dynamical properties of reaction network models, inspired by intuition from chemistry and thermodynamics.
The multidegree of a multiprojective variety encodes intersection number of the variety with different products of linear spaces. Characterizing all possible multidegrees of irreducible varieties is a hard problem (with an almost full classification only in the single and bi-graded case). Here we focus only on the support of the multidegrees which turns out to be a special type of polytope. We'll see some applications to Schubert polynomials and mixed volumes of polytopes. This is joint work with Yairon Cid-Ruiz, Bingling Li, Jonathan Montaño, and Naizhen Zhang.
Two-parameter eigenvalue problems naturally arise when separation of variables is applied to boundary value problems, but also find a variety of other applications. We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices in the initial problem. It is applicable amongst other for a class of Helmholtz equations after separation of variables.
Notions of ranks and border rank abounds in the literature. Polynomials with vanishing hessian and their classification is also a classical problem. Motivated by an observation of Ottaviani, we will discuss why when looking at concise polynomials of minimal border rank, being wild, i.e. their smoothable rank is strictly larger than their border rank, are the same as having vanishing Hessian. The main tool we are using here is the recent work of Buczynska and Buczynski relating the border rank of polynomials and tensors to multigraded Hilbert scheme. From here, we found two infinite series of wild polynomials and we will try to describe their border varieties of sums of powers, which is an analogue of the variety of sums of powers.
The talk is based on joint work with Emanuele Ventura and Mateusz Michalek: https://arxiv.org/pdf/1912.13174.pdf
I will give an introduction to orbital degeneracy loci, a general construction of varieties generalizing zero loci of sections of vector bundles and degeneracy loci of morphisms. The main ingredient is the orbit structure of certain representations of complex Lie groups. As a sample of applications, I will explain how to construct an abelian surface from a skew-symmetric three-form, in the spirit of the classical construction of an elliptic curve from a cubic form.
Many varieties of polynomials carry a canonical action of the general linear group. This talk gives an introduction on how representation theory can be used in the study of the equations of such varieties. We then focus on recent research in geometric complexity theory on continuant orbit closures and plethysm coefficients.
Let X be a smooth projective variety over CC with an action of (CC, +). Assume that X has a unique fixed point x_0. Carrell's conjecture predicts that X is rational. Restriction of orbits to germs at x_0 reduces this conjecture to describing solutions of certain systems of PDE in the formal power series ring $k[[t]] with d(t) = -t^2.$ This suggests a stronger form of the conjecture: X is a union of affine spaces. This strengthening would give an analogue of Bialynicki-Birula decomposition for (CC, +).
In the talk I will explain the beautiful basics on how the (CC, +)-actions, differential equations and rationality intertwine and then present the state of the art on the conjecture. This is a work in progress, comments and suggestions are welcome!
The standard algorithm to compute tropical varieties makes crucial use of the fact that the tropicalization of an irreducible variety is connected. I will discuss joint work with Josephine Yu showing that the tropicalization of a d-dimensional irreducible variety satisfies a stronger d-connectedness property. The proof involves a tropical Bertini theorem.
Minimal submanifolds are mathematical abstractions of soap films: they minimize the Riemannian volume locally around every point. Finding minimal algebraic hypersurfaces in $R^n$ for each n is a long-standing open problem posed by Hsiang. In 2010 Tkachev gave a partial solution to this problem showing that the hypersurface of n x n real matrices of corank one is minimal. I will discuss the following generalization of this fact to all determinantal matrix varieties: for any m, n and r
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.
I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.
Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. In this talk I want to illustrate how these two fields complement each other. The focus lies on the 3264 conics that are tangent to five given conics in the plane. I will illustrate tools and techniques used in numerical algebraic geometry and how we used these to find a fully real instance of this classic problem.
This is joint work with P. Breiding and B. Sturmfels.
A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. It is a classical question to determine whether a given matroid is representable as a vector configuration over a field. Such a matroid is called linear.
This talk addresses generalisations of such representations over division rings or matrix rings which are called skew linear and multilinear matroids respectively.We will describe a generalised Dowling geometry that encodes non commutative equations in matroids. This construction allows us to reduce word problem instances to skew linear or multilinear matroid representations.
The talk is based on joint work with Rudi Pendavingh and Geva Yashfe.
Polynomial functors are like spaces of objects (e.g. k-way tensors) without fixed size and come with an action of (products of) general linear groups. The aim of this talk is to answer the following question: what happens when you replace vector spaces by polynomial functors when defining affine spaces?
I will define polynomial functors, the maps between them and their Zariski-closed subsets and give examples of these things. Then, I will discuss how to extend some of the basic results from affine algebraic geometry to this setting. This is joint work with Jan Draisma, Rob Eggermont and Andrew Snowden.
The analysis of time series is a standard task in data science. Usually, as a first step, features of a time series must be extracted that characterize the series, maybe modulo irrelevant (depending on the application) group actions on the original data. In this talk I will discuss the action of time-warping: the features should be invariant to the speed at which the time-series is run through. This leads, as we show, to quasisymmetric functions, and I discuss their Hopf algebraic setup.
Based on joint work with Kurusch Ebrahimi-Fard and Nikolas Tapia.
Amendola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems admits such a decomposition if and only if the corresponding monodromy group is imprimitive. A consequence of Esterov's classification of sparse polynomial systems with imprimitive monodromy groups is that this decomposition is obtained by inspection. Using these ideas, we present a recursive algorithm to numerically solve decomposable sparse systems. This is joint work with Frank Sottile, Jose Rodriguez, and Thomas Yahl.
Mass-action networks (edge labelled directed graphs) model cascades of chemical reactions (e.g. used by biological systems for adapting to the environment). From the assumption of mass-action kinetics, a mass-action network gives rise to a polynomial dynamical system. In this large class of polynomial systems, the intuition from Chemistry and Algebraic Geometry feed themselves, giving exciting new results. For example, we will discuss complex balanced mass-action networks, which have a natural chemical interpretation and (conjecturally) completely determines the dynamics of the associated systems (called toric dynamical systems). We will introduce "disguised toric systems", which exploit this relationship the other way around: given a dynamical system, can we build a complex balanced mass-action network for it?
(Joint work with Gheorghe Craciun and Miruna-Ştefana Sorea).
For the action of a group on the plane, the group equivalence problem for curves can be stated as: given two curves, decide if they are related by an element of the group. We describe an efficient equality test, using tools from numerical algebraic geometry, to determine (with "probability-one") whether or not two rational maps have the same image up to Zariski closure. Using signature maps, constructed from differential and joint invariants, we apply this test to solve the group equivalence problem for algebraic curves under the linear action of algebraic groups. In this talk I will discuss the equality test and signature maps for algebraic curves, focusing on the action of the complex Euclidean group for our computations and examples. I will present some of our results comparing the sensitivity of different signature maps. This is based on joint work with Tim Duff.
A hierarchical model is realizable by a simplicial complex that describes the dependency relationships among random variables and the number of states of each random variable. Diaconis and Sturmfels have constructed toric ideals that provide useful information about the model. This talk concerns quantitative properties for families of ideals arising from hierarchical models with the same dependency relations and varying number of states. We introduce and study invariant filtrations of such ideals, and their equivariant Hilbert series. A condition that guarantees this multivariate series is a rational function will be presented. The key is to construct finite automata that recognize languages corresponding to invariant filtrations. Lastly, we show that one can similarly prove the rationality of an equivariant Hilbert series for some filtrations of algebras. This is joint work with Uwe Nagel.
We describe connections between invariant theory and maximum likelihood estimation (ML estimation), in the context of matrix normal models. Namely, we link ML estimation in that case to the left right action of SLxSL on tuples of matrices. This enables us to characterize ML estimation by stability under that group action. Furthermore, invariant theory provides a new upper bound on the sample size for generic boundedness of the log-likelihood function. To illuminate the theory the talk puts emphasis on several examples. At the end we briefly outline how our results generalize to Gaussian group models.
Based on joint work with Carlos Améndola, Kathlén Kohn and Anna Seigal. This is the second part of a two part talk: in the first part, Anna Seigal will discuss our results for log-linear models.
We describe connections between invariant theory and maximum likelihood estimation, in the context of log-linear models. Finding a maximum likelihood estimate (MLE) is an optimisation problem over a statistical model, to obtain the point that best fits observed data. We show that this is equivalent to a capacity problem - finding the point of minimal norm in an orbit under a corresponding torus action. The existence of the MLE can then be characterized by stability under the action. Moreover, algorithms from statistics can be used in invariant theory, and vice versa. Based on joint work with Carlos Améndola, Kathlén Kohn and Philipp Reichenbach. This is part one of a two part talk: in the second part, Philipp Reichenbach will discuss our results for multivariate Gaussian models.
A shelling order is a recursive way of constructing a polyhedral complex that helps to understand several topological, algebraic and combinatorial invariants. Consequently, a significant amount of effort has been put into developing techniques to determine if a given complex has a shelling order. In this talk we will explore a different point of view that is less popular: for a complex that admits many shelling orders, a good choice of the shelling order can can make a significant difference. We address this problem for matroid independence complexes, present an intriguing connection with shelling orders of polytopes, and discuss some experiments aimed at better understanding some old problems. This is based on joint work Alex Heaton.
Matroids were introduced by Whitney in 1935 to provide an abstract generalization of the notion of linear independence. Whitney noted that matroids arise naturally from graphs and from matrices. More recently, people have discovered ties to matroid theory and algebraic geometry. In this talk, I will first introduce matroid theory, along with some key examples, and central questions. I will then discuss connections between matroid theory and nonlinear algebra.
In a recent paper by Boij and Conca the upper and lower bounds for the Hilbert function of subalgebras of a polynomial ring are discussed. In this talk we will study subalgebras generated in degree two with minimal Hilbert function. These subalgebras are generated by strongly stable sets of monomials. To minimize the Hilbert function we want to firstly minimize the numbers of variables, and secondly the multiplicity of the algebra. This boils down to a purely combinatorial problem, as the multiplicity can be computed by counting the number of maximal north-east lattice paths in an diagram representing the strongly stable set.
The n-th punctual Hilbert scheme $Hilb^n_0(A^d)$ of points of affine d-space parametrises ideals of finite co-length n of the ring of functions on d-dimensional affine space, whose radical is the maximal ideal at the origin (equivalently, subschemes of length n with support at the origin). A classical theorem of Briancon claims the irreducibility of this space for d=2 and arbitrary n. The case of a small number of points being straightforward, the first nontrivial case is the case of 4 points in 3-space. We show, answering a question of Sturmfels, that over the complex numbers $Hilb^4_0(A^3)$ is irreducible. We use a combination of arguments from computer algebra and representation theory.
A statistical model is a subset of a probability simplex. The maximum likelihood estimation (MLE) is a map that takes an empirical data point and assigns it a point in the model that maximizes the log-likelihood function defined by the data point. Finding the maximum likelihood estimate for a given data point is a problem that can be solved using optimization tools. On the other hand, given a point in the model, we may study the fiber of the MLE map corresponding to that point, known as its logarithmic Voronoi cell. Each logarithmic Voronoi cell lives inside its log-normal polytope, and these log-normal polytopes corresponding to the points of our model fill the probability simplex. We introduce these notions and describe the situation geometrically using examples.
Del Pezzo surfaces are classified by their degree, an integer between 1 and 9. Famous examples are those of degree 3, which are cubic surfaces in $P^3$. In this talk I will focus on del Pezzo surfaces of degree 1. After briefly describing their geometry, I will talk about the set of Q-valued (rational) points on such a surface. I will show what is known about this set so far, and which questions are still open.
Architecture Definition is the process in systems engineering of developing an architectural structure appropriate for achieving a given set of system requirements and constraints. Last winter, I was contacted by a colleague from systems engineering interested in developing a mathematical formalism for defining architecture, classifying architectures and comparing architectures. What emerged uses a combination of model theory from logic and category theory. I will talk about this work, as well as the occasionally surreal experience of collaborating with systems engineers, whom it transpires are even more obsessed with precision of language than mathematicians and walk a line between abstract philosophy and practical engineering.
Geometric rigidity theory is concerned with how much information about a configuration p of n points in a d-dimensional Euclidean space is determined by pairwise Euclidean distance measurements, indexed by the edges of a graph G with n vertices. One can turn this around, and, define, for a fixed graph G, a “measurement variety" associated with all possible edge lengths measurements as the configuration varies. I’ll survey some (somewhat) recent results in geometric rigidity obtained by studying the geometry of measurement varieties.
Grigoriev introduced a theory of tropicalizing differential equations and their formal power series solutions over a trivially valued field. I will describe some generalizations of this theory and what kinds of information they are capable of detecting. In particular, I will explain how to extend to non-trivially valued coefficients and broader classes of series solutions, as well as a variation adapted to studying positive real solutions.
Flag matroids are combinatorial abstractions of flags of linear subspaces, as matroids are of linear subspaces. In this talk, I will review several equivalent descriptions of the ordinary Dressian, which is a parameter space for tropical linear spaces. I will then introduce the flag Dressian as a tropical analogue of the partial flag variety. This work is joint with Madeline Brandt and Chris Eur.
Design of experiments (DoE) for generalized linear models (GLM) is a recurring topic for practitioners and statisticians. We introduce basic notions and tools from GLM and DoE and see that there exists a fruitful connection between DoE and algebraic structures, namely varieties and semi-algebraic sets. This happens because the optimality conditions of experimental designs are often given as algebraic constraints on the parameters of the GLM. We exemplify this approach for the Bradley-Terry paired comparison model, where we compute the exact solutions for the optimal designs for a small example.
Although extremely simple, Gauss' modular arithmetic is a powerful idea in working with the integers. There is a direct application of this idea in algebraic geometry, where one ""reduces"" a variety modulo an integer. A major undertaking in the second half of the previous century linked topological aspects of the original variety to point counts of its reduction. One can go a little further and carry the analytical properties of the original variety over to its reduction. This gives a refined sense about which subvarieties of the reduction lift back up to the original variety. I will report on joint work with Edgar Costa where we implemented this idea, but mostly, I will give a friendly introduction to the basic concepts.
A theta surface in affine 3-space is the zero set of a Riemann theta function. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This is joint work with Daniele Agostini, Julia Struwe and Bernd Sturmfels which offers a new view on this classical topic through the lens of computation.
We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. Joint work with Alexandros Grosdos, Alexander Heaton, Kaie Kubjas, Olga Kuznetsova, and Georgy Scholten.
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. The primary ideals arising in such a decomposition can be characterized in terms of certain differential equations. In this talk we will understand how this characterization works. We will give an explicit algorithm for computing these differential operators that describe a primary ideal, namely Noetherian operators. For some special cases, we will give an alternative representation of the primary ideal by differential equations playing with the join construction. This is a joint work with Yairon Cid-Ruiz and Bernd Sturmfels.
I will introduce Hyperkähler manifolds, a class of manifolds very much studied in complex algebraic geometry; they can be thought as a higher dimensional alanogue of K3 surfaces. I will present a geometrical problem on them, that is the existence of rational curves, i.e. curves birational to a projective line; the problem has a complete solution in the case of K3 surfaces, but much less is known in the Hyperkähler case.
We discuss algebro-geometric properties of the Zariski closure of cyclic matrix groups or semigroups. We will go through several examples and through an implementation in SageMath. This is joint work with Francesco Galuppi.
A well-studied problem in computer vision is ""structure from motion"", where 3D structures and camera poses are reconstructed from given 2D images taken by the unknown cameras. The most classical instance is the 5-point problem: given 2 images of 5 points, the 3D coordinates of the points and the 2 camera poses can be reconstructed. In fact, given 2 generic images of 5 points, this problem has 20 solutions (i.e., 3D coordinates + 2 camera poses) over the complex numbers. Reconstruction problems which have a finite positive number of solutions given generic input images, such as the 5-point problem, are called ""minimal"". These are the most relevant problem instances for practical algorithms, in particular those with a small generic number of solutions. We formally define minimal problems from the point of view of algebraic geometry. Our algebraic techniques lead to a classification of all minimal problems for point-line arrangements completely observed by any number of cameras. We compute their generic number of solutions with symbolic and numerical methods. This is joint work with Timothy Duff, Anton Leykin, and Tomas Pajdla.
Riemann's theta function sits at the intersection of many different fields of mathematics. In this talk, I will present recent results that highlight some of its various aspects: from the Schottky problem in classical geometry to water waves in mathematical physics, passing through numerical algebraic geometry, tropical geometry and statistics.
Fundamental methodology in statistics relies on theoretical and numerical linear algebra. In recent years, nonlinear algebra applied to statistics has fueled the field of algebraic statistics, offering novel interpretations to classical statistical problems. In this talk, we will give an overview of the strong connection between nonlinear algebra and algebraic statistics. In particular, a recent application of algebraic statistics to time series analysis expands the class of statistical models we can study with nonlinear algebraic tools.
Tropical varieties are special rational polyhedral complexes. They are often described as piecewise linear shadows of algebraic varieties from which they are constructed via tropicalization. This process strongly depends on the embedding of the algebraic variety in some algebraic torus. This yields different tropical models of the same variety, and therefore it is important to understand which one reproduces best the original properties. Along these lines we investigate extrinsic and intrinsic tropicalization of cubic surfaces and their 27 lines.
I will report on an algorithm for sampling probability distributions on real algebraic varieties. This algorithm is based on intersecting with linear spaces, produces i.i.d. samples and can handle several connected components. I will further discuss possible extensions of this idea.
Computational algebraic geometry offers a concrete approach to study systems of polynomial equations in several variables. This is useful in several applications in engineering and data science. In this talk we focus on its application in Statistics to study graphical models. We study these models using combinatorics, toric geometry and algebra. Our results provide a more general framework, using staged tree models, to understand the defining equations of widely known graphical models such as decomposable models and discrete Bayesian networks.
The border rank of a tensor generalizes the rank of a matrix and provides a measure of the complexity of the associated bilinear map. We present some recent results on the behaviour of border rank under two different notions of tensor product. The first one, the Segre product, offers a completely geometric framework; the second one, the Kronecker product, is related to the geometry of the matrix multiplication tensor. We highlight similarities and differences between the two settings.
A metric d on the finite set {1,…,n} induces a metric on the (n-1)-dimensional probability simplex which is called Wasserstein distance. This distance is well known to statisticians, and possesses properties which make it appealing to several areas of computer science. Moreover, its unit ball is a polytope, whose combinatorial structure encodes information on the metric d. We study the problem of computing the Wasserstein distance from a point to a statistical model described by polynomial equations. After providing all the necessary definitions, I will discuss the combinatorical and algebraic complexity of this problem, with a special focus on models of independent random variables. This is joint work with Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar and Bernd Sturmfels.
Often a good tactic to approach a challenging problem is to go all the way up to a generic case and then find sufficient conditions for the specialization to keep some of the main features of the former. The procedure depends on taking a dramatic number of variables to allow modifying the given data into a generic shape, and usually receives the name of specialization. This classical method is seemingly due to Kronecker and Krull.
We introduce and study a new notion of specialization that generalizes the classical approach. As applications, we also consider the specialization of rational maps and symmetric and Rees powers of a module.
Our main technical tool is to study the generic freeness of local cohomology modules in a graded setting. Our approach works in a quite unrestrictive setting by only assuming that the coefficient ring is Noetherian, and under additional assumptions (e.g., the coefficient ring is reduced or an integral domain) the results are considerably improved. This is joint work with Marc Chardin and Aron Simis.
This talk is an invitation to operads---powerful mathematical tools---applying to all domains of maths (to any symmetric monoidal category), applied maths and mathematical physics. These universal objects, allow to code operations with multiple inputs.
Operads can be considered in very different flavors, ranging from algebra, topology to geometry. In particular, there exists an algebraic operad, ruling associative algebras, a topological operad (little discs), a geometric operad (moduli spaces of curves). The last one codes Gromov—Witten invariants.
Discovered in the 70’s, operads came back into light in the 90’s, due to algebra (Koszul duality), geometry (moduli spaces of curves) and mathematical physics (theory of quantum spaces), under the impulse of Manin and Kontsevitch.
Computer Vision studies image formation in cameras. In particular, we consider the reconstruction problem: Given images in several cameras of the same scene, triangulate the scene, i.e. find the camera locations and the preimages in the world. There is an algebraic part of this theory that ignores the question whether the reconstruction consists of points that are in front of the camera. We address this issue. This involves semi-algebraic geometry, which we discuss in a projective setup.
This is joint work with Sameer Agarwal, Andrew Pryhuber, and Rekha Thomas.
Steinitz's problem asks whether a triangulated sphere is realizable geometrically as the boundary of a convex polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their creation more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with little success. In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of a novel approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.
Let M be a matrix with nonnegative entries. Its nonnegative rank is the smallest natural number r such that M can be written as a sum of r rank one matrices whose entries are nonnegative. Cohen and Rothblum asked in 1993: Given a non-negative matrix with rational entries, does its non-negative rank over the rational numbers agree with its non-negative rank over the real numbers? After 23 years, two groups (Chistikov et al; Shitov) simultaneously posted papers where they construct matrices that answer this question negatively. Matrices whose nonnegative rank over reals differs from their nonnegative rank over rationals have restricted sets of nonnegative factorizations. This means that they are on the boundary of the set of matrices of nonnegative rank at most r. These boundaries are completely understood for matrices of nonnegative rank at most three. Connection with rigidity theory allows to obtain partial understanding of boundaries for higher nonnegative rank. For nonnegative rank at most three, complete understanding of boundaries allows one to derive semi-algebraic characterizations of these sets.
Motivated by extreme value theory, max-linear graphical models have been recently introduced and studied as an alternative to classical linear structural equation models. Instead of having Gaussian or discrete random variables as nodes in the graph, an important feature of max-linear models is that they support heavy-tailed innovations. In this talk I will give the basic definitions to understand and present max-linear models naturally in the framework of tropical linear algebra, hinting how this can help with the problem of studying conditional independence relations. Joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.
Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree.
The commuting variety is a well-studied object in algebraic geometry whose points are pairs of matrices that commute with one another. In this talk I present a generalization of the commuting variety by using the notion of commuting distance of matrices, which counts how many nonscalar matrices are required to form a commuting chain between two given matrices. I will prove that over any algebraically closed or real closed field, the set of pairs of matrices with bounded commuting distance forms an affine variety. I will also discuss many open problems about these varieties, and present preliminary results in these directions. This is based on joint work with Madeleine Elyze and Alexander Guterman.
