We will consider modeling shapes and fields via topological and lifted-topological transforms. Specifically, we show how the Euler Characteristic Transform and the Lifted Euler Characteristic Transform can be used in practice for statistical analysis of shape and field data. We also state a moduli space of shapes for which we can provide a complexity metric for the shapes. We also provide a sheaf theoretic construction of shape space that does not require diffeomorphisms or correspondence. A direct result of this sheaf theoretic construction is that in three dimensions for meshes, 0-dimensional homology is enough to characterize the shape. We will also discuss Gaussian processes on fiber bundles and applications to evolutionary questions about shapes. Applications in biomedical imaging and evolutionary anthropology will be stated throughout the talk.

Nearfields are easy to define: just omit one of the two distributive laws in the definition of skew fields. We explain the connection of nearfields to sharply 2-transitive permutation groups. Every nearfield has an intrinsic vector space structure, hence one can study nearfields in terms of groups of linear transformations. All finite nearfields have been classified by Zassenhaus in 1936. We report on classification results for infinite nearfields, and we construct some `wild´ nearfields which are far away from skew fields.

In how far is it possible to reconstruct a convex polytope (up to isometry, affine equivalence or combinatorial equivalence) from its edge-graph and some 'graph-compatible metric data', such as its edge-lengths? In this talk we explore this question and pose the following conjecture: a convex polytope P is uniquely determined by its edge-graph, its edge length and the distance of each vertex from some common interior point of P, across all dimensions and combinatorial types. We conjecture even stronger, that two polytopes P and Q are already isometric if they have the same edge-graph, edges in P are at most as long as in Q, and vertex-point distances in P are at least as long as in Q. In other word, a convex polytope cannot become 'bigger' while its edges become shorter. We verify the conjectures in three special cases: if P and Q are combinatorically equivalent, if P and Q are centrally symmetric, and if Q is a slight perturbation of P. These results were obtained using techniques from the intersection of convex geometry and spectral graph theory.

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 describe the bisection fan corresponding to other polyhedral norms. This is joint work with Katharina Jochemko.

Toric bundles are fibre bundles which have a toric variety as a fiber. One particularly studied class of toric bundles are horospherical varieties which are toric bundles over generalized flag varieties. Similar to toric varieties, toric bundles admit a combinatorial description via polyhedral geometry. In my talk, I will explain such a combinatorial description, and describe a couple of results which rely on it. In particular, I will present a generalization of the BKK theorem and the Fano criterion for toric bundles.

The matroid Grassmannian is the space of oriented matroids. 30 years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology. In some easy cases, the matroid Grassmannian is homotopy equivalent to the ordinary real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the ordinary Grassmannians has an analogue on the matroid Grassmannian, namely the monoidal product tof direct sum that is commutative up to all higher homotopies.

Recent developments in tropical geometry and matroid theory have led to the study of “volume polynomials” associated to tropical varieties, the coefficients of which record all possible degrees of top powers of tropical divisors. In this talk, I’ll discuss a volume-theoretic interpretation of volume polynomials of tropical fans; namely, they measure volumes of polyhedral complexes obtained by truncating the tropical fan with normal hyperplanes. I’ll also discuss how this volume-theoretic interpretation inspires a general framework for studying an analogue of the Alexandrov-Fenchel inequalities for degrees of divisors on tropical fans. Parts of this work are joint with Anastasia Nathanson, Lauren Nowak, and Patrick O’Melveny.

In the 1980s Stanley introduced the order polytope and chain polytope associated with a poset while Hibi considered the closely related notions of the Hibi ideal and Hibi ring (as they are now known) associated with a distributive lattice. During the subsequent decades it was realized that these concepts can be applied to the construction and study of toric degenerations of Grassmannians and flag varieties. Moreover, virtually all sufficiently explicit and sufficiently general constructions of such degenerations known today can be interpreted in terms of poset polytopes and Hibi ideals. I will give an overview of these notions and explain how they lead to toric degenerations, reviewing some foundational and some very recent results in this direction.

With Spencer Bloch and Robin de Jong, we recently proved that in a nodal degeneration of smooth curves, the periods of the resulting limit mixed Hodge structure (LMHS) contain arithmetic information. More precisely, if the nodal fiber is identified with a smooth curve C glued at two points p and q then the LMHS relates to the Neron--Tate height of p-q in the Jacobian of C. In making this relation precise, we observed that a "tropical correction term" is required that is based on the degenerate fiber. In this talk, I will explain this circle of ideas with the goal of arriving at the tropical correction term.

Polyhedral Omega is an algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. This synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. After presenting the algorithm, we will see some applications and generalizations.

New types of polyhedral geometry and combinatorial structures emerge in the study of asymptotic geometry of complex manifolds. This includes a multi-scale version of polyhedral geometry, with links to a theory of higher rank combinatorial structures, which combine parts living in different scales at infinity, and a theory of convexity in a piecewise linear setting.
The talk presents some features of these recent developments, and discusses applications in complex geometry.
Based on joint works with Hernan Iriarte, with Noema Nicolussi, and with Matthieu Piquerez.

The well-known greedy algorithm on matroids was adapted by Edmonds to polymatroid polytopes or, essentially equivalent, generalized permutahedra. For a given objective function, the greedy algorithm traces a monotone path in the graph of the polymatroid polytope. The starting point of this work is the observation that this path is precisely the path taken by the shadow-vertex simplex algorithm. It turns out that all monotone paths are greedy paths and all are coherent in the sense of Billera-Sturmfels. Thus, monotone paths are parametrized by the associated monotone path polytope. We show that these monotone path polytope are again a generalized permutahedron and in the case of matroids are the underlying flag matroids. In this talk I will explain the geometry of these “flag polymatroids” and the connection to certain nestohedra associated to lattices of flats via so-called pivot rule polytopes. If time permits, I will also discuss relations to sorting networks and the enumeration of Young tableaux. This is joint work with Alex Black.

Hyperconvexity was introduced by Aronszajn and Panichpadki in the 50', to study nonexpansive mappings between metric spaces. We study a new kind of convexity, defined in terms of lattice properties. We call it ``ambitropical'' as it includes both tropical convexity and its dual, and we relate it with hyperconvexity and mean-payoff games.
Ambitropical convex sets coincide with hyperconvex sets with an additional requirement: stability by an additive group action of the real numbers. They also coincide with the fixed-point sets of Shapley operators, i.e., of dynamic programming operators of (undiscounted) zero-sum games. In this way, ambitropical convex sets provide geometric representations of (calibrated) optimal stationary strategies of mean-payoff games. Moreover, there is a notion of ``ambitropical hull'', unique up to isomorphism, which we construct as the range of a tropical analogue of the Petrov-Galerkin projector, providing an alternative to the ``tight-span'' construction of Isbell and Dress for the injective hull, in the special case of an ambitropical convex set. We finally consider ambitropical polyhedra, defined as ambitropical convex sets that have a cell decomposition in terms of alcoved polyhedra. We relate them with deterministic games and order preserving retracts of the Boolean hypercube.
This talk is based on a joint work with Marianne Akian and Sara Vannucci. A preliminary account of the results appeared in arXiv:2108.07748.

Logarithmic Voronoi cells are convex sets, which arise as fibers of the maximum likelihood estimation map. For discrete models, they live inside their log-normal polytopes, and for certain families of statistical models, the two sets coincide. In particular, logarithmic Voronoi cells for such families are polytopes. I will introduce these notions and investigate when this property holds. I will also talk about how this theory generalizes to Gaussian models, with lots of examples. This talk is based on joint works with Alex Heaton and Serkan Hosten.

The simplex method is one of the most famous and popular algorithms in optimization, it was even named on the top 10 algorithms of the 20th century by SIAM and IEEE. But despite its great success and fame we do not fully understand it. There are several natural open questions arising from the analysis of its performance This talk highlights several fascinating geometric and combinatorial problems we wish to answer. Many open questions will be available for the eager young researcher. I promise!
In the first part I re-introduce natural geometric-topological structure one can associate to the set of all possible monotone paths of a linear program, I will then use this structure, first introduced by Billera and Sturmfels, to study How long can the longest monotone paths on a linear program become? How many different monotone paths can there be on a linear program? We report on two papers, the first joint work with M. Blanchard and Q. Louveaux, and another with C. Athanasiadis and Z. Zhang. Next, the engine of any version of the simplex method is a *pivot rule* that selects the outgoing arc for a current vertex. Pivot rules come in many forms, definitions, and types, but after 80 years we are still ignorant whether there is one that can make the simplex method run in polynomial time. Can we classify all pivot rules? How many possible different pivot rules can there be on a linear Program? Do these questions even make sense? In the second half of my talk will present two recent positive results: For 0/1 polytopes there are explicit pivot rules for which the number of non-degenerate pivots is polynomial and even linear (joint work with A. Black, S. Kafer, L. Sanita). I also present a parametric analysis for all pívot rules. We construct a polytope, the pivot rule polytope, that parametrizes all memoryless pívot rules of a given LP. Its construction is a generalization of the Fiber polytope construction of Billera Sturmfels. This is an attempt to get a complete picture of the structure “space of all pivot rules of an LP” (joint work with A. Black, N.Lütjeharms, and R. Sanyal).

Valuations on polytopes are maps that combine the geometry of polytopes with relations in a group. It turns out that many important invariants of matroids are valuative on the collection of matroid base polytopes, e.g., the Tutte polynomial and its specializations or the Hilbert–Poincaré series of the Chow ring of a matroid. In this talk I will present a framework that allows us to compute such invariants on large classes of matroids, e.g., sparse paving and elementary split matroids, explicitly. The concept of split matroids introduced by Joswig and myself is relatively new. However, this class appears naturally in this context. Moreover, (sparse) paving matroids are split. I will demonstrate the framework by looking at Ehrhart polynomials and Hilbert–Poincaré series of elementary split matroids. This talk is based on the preprint `Valuative invariants for large classes of matroids' which is joint work with Luis Ferroni.

Studies on homogeneous polynomials with the half-plane property were initially motivated by the physical theory of electrical networks. They later became of interest in convex optimization. In particular, a homogeneous polynomial with the half-plane property is hyperbolic with respect to every point in the positive orthant, having an associated hyperbolicity cone. Feasible sets of semi-definite programming, i.e., spectrahedral cones, are hyperbolicity cones for some polynomials. For the converse, the generalized Lax conjecture states that every hyperbolicity cone is spectrahedral.
Considering that the basis generating polynomials of matroids are homogeneous and multiaffine, a natural approach is investigating the mentioned properties of matroids. For instance, the combinatorial structure of matroids was used by Brändén to give a counter example for a stronger version of the conjecture.
In this talk, we present our results on the operations on matroids that preserve related properties. Moreover, we give an algorithm for testing the half-plane property of matroids that uses Macaulay2 packages SumsOfSquares, Matroids and the Julia package Homotopy Continuation.jl. We illustrate the outcome of the classification of matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids on 9 elements.
This is joint work with Mario Kummer.

Every so often, we find a family of simplicial spheres with interesting combinatorial properties, and would like to know if they are realizable as boundaries of convex polytopes. For instance, this happened recently to Zheng; to Novik and Zheng; and to Criado and Santos.
We focus on algorithmically *disproving* the convex realizability of a given simplicial sphere. All previous successful approaches made essential use of integer or linear programming, which severely limits the practical applicability due to memory and runtime constraints. In our new implementation, we instead perform a combinatorial search using optimized data structures, which lets us decide in seconds examples that are completely out of reach of the best alternative implementation by Gouveia, Macchia, and Wiebe.
Our code has now undergone the third complete rewrite, and is almost ready to be merged into the polymake tree. If time permits, we will discuss some of the lessons learned during the implementation, and mention further conceptual improvements that bring us tantalizingly close to deciding the long-standing open problem of convex realizability of multitriangulations.

For a separable binary form of degree n over a complete non-archimedean field, there is a canonical metric tree on n leaves associated to it. Furthermore, for every degree n, there are only finitely many tree types, which gives rise to a partition of the space of all non-archimedean binary forms of degree n.
In this talk, we will focus on the particular case where n = 5. Then, there are exactly 3 unmarked and 5 marked tree types. We will give a set of tropical invariants, valuations of certain elements coming from invariant theory for binary quintics, and show that these invariants allow us to distinguish the tree types algorithmically. As an application, we will express the reduction types of Picard curves in terms of tropical invariants of the associated binary quintics.
This is joint work with Yassine El Maazouz and Paul Alexander Helminck.

The timetable is the core of every public transportation system. The standard mathematical tool for optimizing periodic timetabling problems is the Periodic Event Scheduling Problem (PESP). A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. Finally, we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis in terms of the number of spanning trees.

Tropical geometry is a powerful tool that allows one to compute enumerative algebraic invariants through the use of some correspondence theorem, transforming an algebraic problem into a combinatorial problem. Moreover, the tropical approach also allows one to twist definitions to introduce mysterious refined invariants, obtained by counting tropical curves with polynomial multiplicities. So far, this correspondence has mainly been implemented in toric varieties. In this talk we will study enumeration of curves in abelian surfaces and use the tropical geometry approach to prove a multiple cover formula that enables an simple and elegant computation of enumerative invariants of abelian surfaces.

Symmetric edge polytopes (SEPs) are special lattice polytopes constructed from a simple graph. They play a role in physics, in the study of metric spaces and optimal transport. We study a numerical invariant of symmetric edge polytopes, called the gamma-vector, whose entries are linear combinations of the \(h^*\) numbers. In particular, we target a conjecture of Ohsugi and Tsuchiya which predicts nonnegativity of the gamma-vector of every SEP. In a joint work with Alessio D'Alì, Martina Juhnke-Kubitzke and Daniel Köhne, we prove nonnegativity for one of the entries of the gamma-vector, and descriube the graphs whose SEP attains equality. Moreover, we consider random graphs in the Erdős–Rényi model, and we obtain asymptotic nonnegativity of the whole gamma-vector in certain regimes.

The moduli space of smooth cubic hypersurfaces in \(\mathbb{P}^n\) is an open subset of a projective space. We construct a compactification of the latter which allows us to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points. We term it a \(1\)-complete variety of cubic hypersurfaces in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. Finally, we arrive at the desired numbers in the case of cubic surfaces. This is joint work with Alessandro Danelon, Claudia Fevola, Andreas Kretschmer.

Let \(V=\binom{[n]}{2}\) label the possible diagonals among the vertices of a convex \(n\)-gon. A subset of size \(k+1\) is called a \((k+1)\)-crossing if all elements mutually cross, and a general subset is called \((k+1)\)-crossing free if it does not contain a \(k\)-crossing. \((k+1)\)-crossing free subsets form a simplicial complex that we call the \(k\)-associahedron and denote \(Ass_k{n}\) since for \(k=1\) one (essentially) recovers the simplicial associahedron. The \(k\)-associahedron on the \(n\)-gon is known to be (essentially ) a shellable sphere of dimension \(k(n-2k-1)\) and conjectured to be polytopal (Jonsson 2003). It is also a subword complex in the root system of the A.
The Pfaffian of an anti-symmetric matrix of size \(2k+2\) is the square root of its determinant, and it is a homogeneous polynomial of degree \(k+1\) with one monomial for each possible complete matching among \(2k+2\) nodes representing the rows and columns. Thus, monomials correspond to certain \((k+1)\)-subsets of \(V\) and among them there is a unique \((k+1)\)-crossing. Calling \(I_k(n)\) the ideal of all Pfaffians of degree \(k+1\) in an antisymmetric matrix of size \(n\), it is known (Jonsson and Welker 2007) that for certain term orders the corresponding initial ideal equals the Stanley-Reisner ideal of the \(k\)-associahedron.
In this talk we explore the relation between Pfaffians and the \(k\)-associahedron from the tropical perspective. We show that the part of the tropical Pfaffian variety \(trop(I_k(n))\) lying in the ``four-point positive orthant’’ realises the \(k\)-associahedron as a fan, and that this intersection is contained in (but is not equal to, except for \(k=1\)) the totally positive tropical Pfaffian variety \(trop^+(I_k(n))\). We hope this to be a step towards realising the \(k\)-associahedron as a complete fan, but have only attained this for \(k=1\): we show that for any seed triangulation \(T\), the projection of \(trop^+(I_1(n))\) to the coordinates corresponding to diagonals in T produces a complete polytopal simplicial fan, that is, the normal fan of an associahedron. In fact, the fans we obtain are linearly isomorphic to the \(g\)-vector fans in cluster algebras of type \(A\), as realized by Hoheweg-Pilaud-Stella (2018).

A well known result relates matroid subdivisions of hypersimplices to tropical linear spaces. We continue the study of subdivisions of generalized permutahedra into cells that are generalized permutahedra. This talk will focus on subdivisions of the regular permutahedron into cells that correspond to flag matroids and intervals in a partial ordering on the symmetric group. This is joint work with Michael Joswig, Georg Loho, and Jorge Olarte

SageMath is an open-source general-purpose mathematical system based on Python that integrates computer algebra facilities and general computational packages. Sage, initially developed by Stein and first released in 2005, in over 1.5 decades of incubation in its pseudo-distribution comprising over 300 software packages, has grown to provide over 500 Cython and over 2000 of its own Python modules, ranging from sage.algebras.* over sage.geometry.* to sage.tensor.*, with a total of over 2.2 million lines of code.
I will give an outline of a project, ongoing since 2020, to modularize the Sage library so that parts of it can be flexibly installed and used by other Python projects. One of the first products of the project is the modularized distribution sagemath-polyhedra, which provides computations and constructions with convex polyhedra with multiple backends, including PPL, Normaliz, and polymake.
The project to modularize the Sage library does not conflict with the software integration goals of SageMath. As an example of progress on the latter, I will show the integration of sage.geometry.polyhedron and sage.manifolds, introduced in Sage 9.4.

The braid groups $B_n$ were introduced in the 1930s in the work of Artin. An element in the braid group $B_n$ is called an $n$-braid. Alexander related braids to knots and links in 3-dimensional space, by means of a closure operation. In that realm, it became important to understand the braid representatives of a given link. Markov's theorem relates these representatives by two moves, conjugacy in the braid group, and (de)stabilization, which passes between braid groups. Markov's moves and braid group algebra have become fundamental in Jones' pioneering work and its later continuation towards quantum invariants. Conjugacy is, starting with Garside's, and later many others' work, now relatively well group-theoretically understood. In contrast, the effect of (de)stabilization on conjugacy classes of braid representatives of a given link is in general difficult to control. Only in very special situations can these conjugacy classes be well described.
We are concerned with the question when infinitely many conjugacy classes of $n$-braid representatives of a given link occur. Birman and Menasco introduced a move called exchange move, and proved that it necessarily underlies the switch between many conjugacy classes of $n$-braid representatives of a given link.
After some very brief general introduction to knots and braids and their applications, we will discuss several results when the exchange move is also sufficient for generating infinitely many conjugacy classes of braid representatives.

We present a polymake extension for performing computations in the intersection ring of matroids. Addition in the ring is carried out on the indicator vectors of maximal chains of flats while the product is the usual matroid intersection when the intersection is loopfree and zero otherwise. The primary feature of this extension is the ability to compute the homology groups of the intersection ring induced by deletion and contraction as well as those of any subring generated by a minor-closed class of matroids.

After a brief overview about torsion in homology and examples of complexes with small torsion and few vertices we will focus our attention on a particular series of examples. In particular, we will construct a series HMT(n) of 2-dimensional simplicial complexes with huge torsion, |H_1(HMT(n))|=|det(H(n))|=n^(n/2), where the construction is based on the Hadamard matrices H(n) and they have linearly many vertices in n. Our explicit series improves a previous construction by Speyer, narrowing the gap to the highest possible asymptotic torsion growth achieved by Newman via a randomized construction.

We extend the theory of Lorentzian polynomials to convex cones. This is used to give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid, and reprove the Hodge-Riemann relations of degree one for the Chow ring of a matroid. This is joint work with Jonathan Leake.

Given a convex polytope P and a projection map it is possible to construct the associated fiber polytope, that intuitively is the average of the fibers of P under this projection and has interesting combinatorial properties. In this joint work with Léo Mathis we study this construction for all convex bodies. The aim is to relate aspects of the original convex body with those of its fiber body. I will discuss theoretical results as well as some concrete examples.

Fix non-negative integers g and n such that 2g-2+n>0, the tropical moduli space \Delta_{g,n} is a topological space that parametrizes isomorphism classes of n-marked stable tropical curves of genus g with total volume 1. These spaces are important because their reduced rational homology is identified equivariantly with the top weight cohomology of the algebraic moduli spaces M_{g,n}, with respect to the S_n-action of permuting marked points. In this talk, we focus on the genus 2 case and compute the characters of \tilde{H}_i(\Delta_{2,n};Q) as S_n-representations for n up to 8. To carry out the computations, we use a cellular chain complex for symmetric \Delta-complexes, a technique developed by Chan, Galatius, and Payne. We will also briefly mention work in progress that extends the computations to n=10, in collaboration with Christin Bibby, Melody Chan, and Nir Gadish. All computations are done in SageMath.

We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyperplane arrangements with applications to physics and computer science. This is joint work with Taylor Brysiewicz and Lukas Kuehne.

We contribute to a better understanding of the class of functions that is represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning tasks. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). We also present upper bounds on the sizes of neural networks required for exact function representation.

Based on analogies between algebraic curves and graphs, there are several graph parameters defined. In this talk we will study the so-called stable gonality. The stable gonality is a measure for the complexity of a graph and is defined using morphisms from a graph to a tree. We show that computing the stable gonality of a graph is NP-hard. This is joint work with Dion Gijswijt and Harry Smit.

We identify the scattering equations from particle physics as the likelihood equations for a particular statistical model. The scattering potential plays the role of the log-likelihood function. We employ recent methods from numerical nonlinear algebra to solve challenging instances of the scattering equations. We revisit the theory of stringy canonical forms proposed by Arkani-Hamed, He and Lam, introducing positive statistical models and their amplitudes. This is joint work with Bernd Sturmfels.

The Hyperdeterminant is a celebrated tool in tensor theory, geometrically it represents the discriminant of a Segre embedding. The Segre embedding is a toric map associated to a Cayley sum, probably the basic example of Cayley polytopes. Unlike other discriminants the hyperdeterminant is at times accessible to computation due to a decomposition method developed by Schäfli. We propose a generalisation of the Schäfli method to discriminants associated to general Cayley sums and show how this can be treated as a discriminant of a system of polynomials. This is joint work with A. Dickenstein and R. Morrison.

A Conjecture by Conway and Sloane from the 90s can be phrased as ''Every tropical abelian variety is determined by its tropical theta constants''. It was proved by Conway and Sloane in dimension up to 3 and by Vallentin in dimension 4. We prove it in dimension 5. The prove relies on the classification of iso-edge domains in dimension 5: These are a variant of the iso-Delaunay decomposition introduced by Baranovskii and Ryshkov. We further characterize the so-called matroidal locus in terms of the tropical theta constants in dimension up to 5 and point out connections to the tropical Schottky problem. The talk is based on a joint work with Mathieu Dutour Sikirić.

The higher Stasheff–Tamari orders are two partial orders introduced in the 1990s by Kapranov, Voevodsky, Edelman and Reiner on the set of triangulations of a cyclic polytope. Edelman and Reiner conjectured these two orders to coincide, which remains an open problem today. In this talk we give new combinatorial interpretations of the orders. These interpretations differ depending on whether the cyclic polytope is 2d-dimensional or (2d + 1)-dimensional, but are expressed in each case in terms of the d-skeleton of the triangulation. This naturally leads us to also characterise the d-skeletons of triangulations of (2d + 1)-dimensional cyclic polytopes, complementing the description of the d-skeleton of a 2d-dimensional triangulation given by Oppermann and Thomas. Our results make the conjectured equivalence of the orders a more tractable problem, and we use them to run computational experiments checking the conjecture, to which we find no counter-examples. Our results also have interpretations in the representation theory of algebras, but we will not discuss these connections.

In the 90's the study of visibility graphs prompted the definition of "persistent graphs", a class of vertex-ordered graphs defined by three simple combinatorial properties. We reveal a surprising connection of these persistent graphs to the triangulations of the 3-dimensional cyclic polytope via a natural bijection.
(Joint work with Vincent Froese)

Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of mass-action kinetics. Various questions concerning the signs of multivariate polynomials in positive orthant arise from studying the solutions' qualitative behavior with respect to parameter values, in particular the existence of multistationarity. In this work, we revisit a method of detecting multistationarity, in which we utilize the circuit polynomials to find symbolic certificates of nonnegativity for 2-site phosphorylation cycle in a recent work of the speaker joint with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff. Moreover, we will give a description of all possible certificates that can arise from 2-site phosphorylation cycle, and provide further insight into the number of positive steady states of the n-site phosphorylation cycle model.

In this talk, we look at the problem of minimizing the number of critical simplices (Min-Morse) from the point of view of inapproximability and parameterized complexity. Let n denote the size of a simplicial complex. We first show that Min-Morse can not be approximated within a factor of 2^{\log^{(1-\epsilon)}n}- \delta, for any \epsilon,\delta>0, unless NP \subseteq QP. Our second result shows that Min-Morse is WP-hard with respect to the standard parameter. Next, we show that Min-Morse with standard parameterization has no FPT-approximation algorithm for any approximation factor \rho, unless WP = FPT. Since the gadgets involved in our reduction are 2-complexes, the above hardness results are applicable for all complexes of dimension \geq 2.

Donaldson-Thomas theory extracts numerical invariants from an algebraic variety X by examining the geometry of the Hilbert schemes of X. While the sister subject of Gromov-Witten theory has had a long and rich interaction with tropical and polyhedral methods, the relationship between the Hilbert scheme and tropical geometry is less developed. I will give an introduction to tropical and logarithmic aspects of Donaldson-Thomas theory, from the point of view of the GKZ secondary polytope construction, and generalizations thereof. This is based on joint work with Davesh Maulik (MIT).

I'll give an introduction to the moduli space of tropical abelian varieties, assuming no background in tropical geometry. Lots of different combinatorics arises, including the beautiful century-old combinatorics of Voronoi reduction theory, perfect quadratic forms, regular matroids, and metric graphs. On the geometric side, it relates to toroidal compactifications of the classical moduli space A_g of abelian varieties. I'll explain how, and, time permitting, I'll report on work-in-progress in which we use tropical techniques to find new rational cohomology classes in A_g in a previously inaccessible range. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

This talk deals with two fundamental objects of discrete mathematics that are closely related - (convex) polytopes and matroids. Both appear in many areas of mathematics, e.g., algebraic geometry, topology and optimization.
A classical question in polyhedral combinatorics is 'Does the vertex-edge graph of a d dimensional polytope determine its face lattice?'. In general the answer is no, but a famous result of Blind and Mani, and later Kalai is a positive answer of that question for simple polytopes. In my talk I discuss this reconstructability question for the special class of matroid (base) polytopes. Matroids encode an abstract version of dependency and independency, and thus generalize graphs, point configurations in vector spaces and algebraic extensions of fields. Maximal independent sets in a matroid satisfy a basis-exchange axiom. The exchanges correspond to edges in the basis-exchange graph which is the vertex-edge graph of the matroid polytope.
This is joint work with Guillermo Pineda-Villavicencio

Stacky fans are orbifold objects in discrete geometry with a rich combinatorial structure. They arise naturally as the moduli spaces of tropical curves. Explicitly, a stacky fan is built from polyhedral cones quotiented out by finite groups.
We study stacky fans embedded in others and offer algorithms to compute their relative homology. We then apply this to specific loci of tropically smooth plane curves embedded in the moduli of genus 3 tropical curves. This is joint ongoing work with Michael Joswig.

Given a polytope P we study the set of all possible Minkowski summands. Being a Minkowski summand is equivalent to normal (or strongly combinatorial) equivalence. Different points of views provide different looking (but equivalent) parametrizations of the space of Minkowski summands as a polyhedral cone. We will go over several examples like polygons, permutohedra, cubes, and more.

We consider a general combinatorial framework for constructing mirrors of quasi-smooth Calabi-Yau hypersurfaces defined by weighted homogeneous polynomials. Our mirror construction shows how to obtain mirrors being Calabi-Yau compactifications of non-degenerate affine hypersurfaces associated to certain Newton polytopes. This talk is based on joint work with Victor Batyrev.

A polytrope is a convex polytope that is expressed as the tropical convex hull of a finite number of points. It is well known that every bounded cell of a tropical linear space is a polytrope, and its converse statement has been a conjecture. We develop an elementary approach to the relationship between tropical convexity and tropical linearity, and show that the conjecture holds in dimension up to 3 and fails in every higher dimension. Thus, tropical convexity is strictly bigger than tropical linearity.

We portray the evolution of the Covid-19 epidemic during the crisis of March-April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from Perron-Frobenius theory and tropical geometry.

Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first counter examples and Rivin gave a complete resolution. In dimensions 4 and up, Mnev's universality theorem renders the question for inscribable combinatorial types hopeless. In this talk, I will address the following refined question: Given a polytope P, is there a normally equivalent polytope with vertices on a sphere? That is, can P be deformed into an inscribed polytope while preserving its normal fan? It turns out that the answer gives a rich interplay of geometry and combinatorics, involving local reflections and type cones. This is based on joint work with Sebastian Manecke (who will continue the story in the FU discrete geometry seminar next week).

In the geometry of polynomials, the notion of stability is of prominent importance. The purpose of the talk is to discuss its recent generalization to conic stability. Given a convex cone K in real n-space, a multivariate polynomial f in C[z] is called K-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the non-negative orthant, then $K$-stability specializes to the usual notion of stability of polynomials. In particular, we focus on $K$-stability with respect to the positive semidefinite cone and develop stability criteria building upon the connection to the containment problem for spectrahedra, to positive maps and to determinantal representations. These results are based on joint work with Papri Dey and Stephan Gardoll.

Classically, there is a rich theory in algebraic combinatorics surrounding the various objects associated with a generic real matrix. Examples include regular triangulations of the product of two simplices, coherent matching fields, and realizable oriented matroids. In this talk, we will extend the theory by skipping the matrix and starting with an arbitrary triangulation of the product of two simplices instead. In particular, we show that every polyhedral matching field induces oriented matroids. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex. Furthermore, we give a corresponding topological construction using Viro’s patchworking. This talk will also sketch the relationship between Baker-Bowler’s matroids over hyperfields and our work. This is joint work with Marcel Celaya and Chi-Ho Yuen.

In this talk I will describe very recent work on analyzing different models of auto-encoder neural networks using random polytopes. In general, a class of neural networks known as auto-encoders can be understood as a low- dimensional (piecewise linear) embedding ℝᴺ → ℝⁿ (with N >> n), where points in the domain are e.g. images (in their pixel form) and each dimension the range extracts a single "feature".
Although understanding the properties of these embedings is crucial for explainability (e.g. interpreting the decisions taken by the network), the research in this direction has been undertaken only from the statistical point of view. We introduce `random polytope descriptors` which try to approximate datasets in the feature space by possibly tight convex bodies. This is the first attempt at understanding the very coarse geometry of embeddings provided by neural networks. Using such descriptors one can answer e.g. if the natural clusters of of images has been preserved or distorted by the neural network, and if the embedding actually separates them (the question of entanglement).
We use random polytope descriptors to examine the behaviour of auto-encoder networks in both vanilla and variational form, and provide first evidence for susseptibility of the latter to out-of-distribution attacks.
This is joint work with L.Ruffs (TUB, Department of Electrical Engineering and Computer Science) and M.Joswig (TUB, Chair of Discrete Mathematics/Geometry; MPI for Mathematics in the Sciences, Leipzig).

Many fundamental topological problems about 3-manifolds are algorithmically solvable in theory, but continue to withstand practical computations. In recent years some of these problems have been shown to allow efficient solutions, as long as the input 3-manifold comes with a sufficiently "thin" presentation.
More specifically, a 3-manifold given as a triangulation is considered thin, if the treewidth of its dual graph is small. I will show how this combinatorial parameter, defined on a triangulation, can be linked back to purely topological properties of the underlying manifold. From this connection it can then be followed that, for some 3-manifolds, we cannot hope for a thin triangulation.
This is joint work with Kristóf Huszár and Uli Wagner

A discriminant is a subset of a function space consisting of singular functions. We shall consider discriminants of finite dimensional vector spaces of polynomials defining hypersurfaces in a toric variety. In the best case scenario (for plane curves for example), the discriminant is an irreducible hypersurface, however this is not the case for almost any other toric variety. We discuss positive results on the geometry of discriminants of weighted projective spaces and how to compute them via the A-discriminants of Gelfand, Kapranov and Zelevinsky.

I will report on the implementation of hyperplane arrangements in polymake. Hyperplane arrangements appear in a wide variety of applications in tropical and algebraic geometry. An important construction is the induced subdivision. We will discuss the new HyperplaneArrangement object, its properties and our simple algorithm for constructing the cell subdivision.
This is joint work with Marta Panizzut.

We consider a model for generating non-simple, non-simplicial random polytopes. The first step of the random process generates a random polytope P via random hyperplanes tangent to the d-dimensional sphere; the second step is given by the convex hull of a binomial sample of the vertices of P. In this talk we will discuss some ongoing work establishing results about the expected complexity of such polytopes.

Tropically planar graphs are graphs that arise as the skeletons of smooth tropical plane curves, which are dual to regular unimodular triangulations of lattice polygons. In this talk we study what geometric properties such graphs have in the moduli space of metric graphs, as well as the combinatorial properties these graphs satisfy. This talk will include older work with Brodsky, Joswig, and Sturmfels; newer work with Coles, Dutta, Jiang, and Scharf; and ongoing work with Tewari.

In this talk I will present a model for the realization space of a polytope which represents a polytope by its slack matrix. This model provides a natural algebraic relaxation for the realization space, and comes with a defining ideal which can be used as a computational engine to answer questions about the realization space. We will see how this model is related to more classical realization space models (representing realizations by Gale diagrams or points of the Grassmannian). In particular, we will see these relationships can be used to improve computational efficiency of the slack model.

In 2010, Stapledon described a generalization of Ehrhart theory with group actions. In 2018, Ardila, Schindler, and I made progress towards answering one of Stapledon's open problems that asked to determine the equivariant Ehrhart theory of the permutahedron. We proved some general results about the fixed polytopes of the permutahedron, which are the polytopes that are fixed by acting on the permutahedron by a permutation. In particular, we computed their dimension, showed that they are combinatorially equivalent to permutahedra, provided hyperplane and vertex descriptions, and proved that they are zonotopes. Lastly, we obtained a formula for the volume of these fixed polytopes, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. Building off of the work of the aforementioned, we determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This project presents combinatorial formulas for the Ehrhart quasipolynomials and Ehrhart Series of the fixed polytopes of the permutahedron, along with other results regarding interpretations of the equivariant analogue of the Ehrhart series. This is joint work with Federico Ardila (San Francisco State University) and Mariel Supina (UC Berkeley).