Contributed talks

Viktoriia Borovik (Osnabrück University) - Toric degenerations of Grassmannian parametrizations

There are many different combinatorial approaches to constructing toric degenerations of the Grassmannian. These constructions arise from: representation theory, such as degenerations associated to chain-order polytopes of the Young poset, or via matching fields in the sense of Sturmfels and Zelevinsky. We consider a birational parametrization of the Plücker embedding and study the graph of this map. The question we pose is: can we lift some toric degeneration of the Grassmannian to this graph? 

Danai Deligeorgaki (KTH) - On the marginal independence structure of DAG models

We consider the problem of estimating the marginal independence structure of a DAG model from observational data. In order to so, we divide the space of directed acyclic graphs (DAGs) into certain equivalence classes, where each class can be represented by a unique undirected graph called the unconditional dependence graph. The unconditional dependence graphs satisfy certain graphical properties, namely having equal intersection and independence number. Using this observation, we can construct a Grobner basis for an associated toric ideal and define additional binomial relations to connect the space of unconditional dependence graphs. With these moves, we can implement a search algorithm, GrUES (Grobner-based Unconditional Equivalence Search), that estimates the conditional independence structure of the graphical model. The implementation shows that GrUES recovers the true marginal independence structure via a BIC-optimal or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior. This is joint work with Alex Markham, Pratik Misra and Liam Solus.

Kevin Kühn, Arne Kuhrs (Goethe University Frankfurt) - Buildings, tropical linear spaces and (valuated) matroids - with a view towards a real version

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne's result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid, extending a result of Dress and Terhalle. This is joint work with Luca Battistella, Martin Ulirsch and Alejandro Vargas.

We will also present ongoing work about a real analogue of the story which was drawn to our attention by Kris Shaw at the last graduate student meeting in Stockholm. We show that a signed version of the Goldman-Iwahori space is the limit of all real tropicalizations of linear spaces over real closed fields, i.e. of real Bergman fans of (realizable) oriented matroids. 

Nataliia Kuschnerchuk (Aalto University) - Matroid stratification of ML degrees of independence models

We study the maximum likelihood (ML) degree of discrete exponential independence models and models defined by the second hypersimplex. For models with two independent variables, we show that the ML degree is an invariant of a matroid associated to the model. We use this description to explore ML degrees via hyperplane arrangements. For independence models with more variables, we investigate the connection between the vanishing of factors of its principal A-determinant and its ML degree. Similarly, for models defined by the second hypersimplex, we determine its principal A-determinant and give computational evidence towards a conjectured lower bound of its ML degree.

Emiliano Liwski (KU Luevenb) - Paving Matroids: Defining Equations and Associated Varieties

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we present two distinct methods for generating polynomials within the associated ideals of these varieties across any dimension. Additionally, we explain the relationship between polynomials constructed using these different methods. We then compute a comprehensive and finite set of defining equations for matroid varieties associated with specific classes of paving matroids. Lastly, we present several examples applying our results and compare them with the known cases in the literature. This is a joint work with Fatemeh Mohammadi. 

Teemu Lundström (Aalto University) - f-vector inequalities for order and chain polytopes

From a finite poset P one can construct the so called order and chain polytopes, denoted by O(P) and C(P) respectively. These are 0/1-polytopes where both the vertices and facets are easily described as corresponding to certain parts of the underlying poset. This talk will be focused on the f-vectors on these polytopes. In 2016, it was proved by Hibi and Li that if a poset P does not have a certain X-shaped poset as a subposet, then O(P) and C(P) are unimodularly equivalent, and hence have the same f-vectors. In the same paper, Hibi and Li conjectured that for any poset P the f-vector of C(P) coordinate-wise dominates the f-vector of O(P). Some special cases of this conjecture are known but the full conjecture remains open. In this talk, I will go over some of the basic properties of order and chain polytopes and what is known about them. I will present our work that takes a new step towards proving the conjecture raised by Hibi and Li. Our main theorem is that the inequality conjectured by Hibi and Li holds for all posets P in a family of posets built by taking disjoint and connected unions, starting with all posets lacking the X-shaped poset as subposet. This is joint work with Ragnar Freij-Hollanti. 

Victoria Schleis  (Universität Tübingen/Durham University) - Quiver polytopes and their subdivisions

Matroid polytopes and their matroid subdivisions are not just combinatorially interesting objects: they also play a significant role in tropical geometry. They equivalently describe tropical linear spaces, and provide a way to study valuated matroids geometrically. In recent joint works with Alessio Borzi and Giulia Iezzi, we introduced tropical and matroidal analogues for quiver Grassmannians, which are combinatorially rich generalizations of flag varieties. In this talk, I will discuss some of my current work in progress, describing polyhedral analogues of these concepts. In the process, I will give an introduction to Coxeter matroid polytopes, their subdivisions and their importance in matroid theory and tropical geometry.

Maximilian Wiesmann (MPI MiS Leipzig) - Geometry of Polynomial Neural Networks

In this talk, we present results about the expressivity and learning process for polynomial neural networks (PNNs) with monomial activation functions. The weights of the network parametrize the neuromanifold. We study neuromanifolds using tools from algebraic geometry: we give explicit descriptions as semialgebraic sets and characterize their Zariski closures, called neurovarieties. We study their dimension and associate an algebraic degree, the learning degree, to the neurovariety. The dimension serves as a geometric measure for the expressivity of the network, the learning degree is a measure for the complexity of training the network and provides upper bounds on the number of learnable functions.

Fiona Young (Cornell University) - The essential bound of a k-polymatroid and applications to excluded minor problems

The singleton and doubleton minors of a polymatroid encode a surprising amount of information about its structural complexity. Starting with a $k$-polymatroid $\rho$, we subtract from it as many maximally-separated matroids as possible. Let the result be an $m$-polymatroid; this gives rise to a notion of boundedness for $\rho$. When $k$ is sufficiently large, the bounds on the singleton and doubleton minors of $\rho$ completely determine the bound on $\rho$. Much of this is motivated and guided by the polytopal perspective of polymatroids. Our results provide an organized framework for thinking about polymatroid excluded minor problems.

Bailee Zacovic (University of Bonn) - The matroid complex and its homology

A matroid is a combinatorial object which abstracts notions of independence appearing across mathematics. As a part of his work on non-commutative symplectic geometry, Kontsevich defines the graph complex, which gives rise to a matroid analogue. Recently, the homology of the graph complex was identified with the reduced rational homology of spaces of stable tropical curves and the top weight cohomology of moduli spaces of algebraic curves, generating interest in the homology of the matroid complex. By extending the Macaulay2 Matroids package, we compute the rational matroid complex in small cases via a decomposition into fixed rank-subcomplexes, and discover no homology in degrees $\leq 8.$ We also extend recent results pertaining to odd wheels in the graph complex, as well as prove more general results for rank-$2$ matroids. This work is ongoing and joint with Juliette Bruce, Benjamin Ashlock, and Jacob Bucciarelli.

Software demonstrations

Oskar Henriksson (University of Copenhagen) - Tropical root bounds and generalized polyhedral start systems in OSCAR

A key step in solving a polynomial system with homotopy continuation is to find an upper bound on the number of solutions, as well as easy-to-solve start systems with precisely that many solutions. In this presentation, I’ll demonstrate a new Julia package based on OSCAR for computing root bounds and associated start systems, based on the idea of sharpening Bernstein’s mixed volume bound, by replacing it with a certain tropical intersection number. The package builds on (and in some cases extends) existing functionality for tropical geometry in OSCAR, and outputs homotopies that can be used directly by HomotopyContinuation.jl. As a case study, we use steady state systems from reaction network theory, where one can show that the improved tropical root bounds are generically sharp.

This is a combination of several joint works in progress with Elisenda Feliu, Paul Helminck, Yue Ren, Benjamin Schröter and Máté Telek.

Lena Weis (TU Berlin) - Tropical geometry in polymake

Through interactive visualization tools and practical examples, we will gain insight into the interplay between combinatorial and algebraic aspects inherent in tropical geometry. From exploring tropical polytopes to understanding tropical varieties, this demonstration provides a hands-on experience that illuminates significance of tropical geometry within a computational framework.