Nicolas Anderson - Queen Mary University London
Title: A Closure Operation for Valuated Matroids
Abstract: We introduce a new cryptomorphism for valuated matroids in terms of a closure operation, which we call a valuated closure operator. Given a totally ordered idempotent semifield R, a valuated closure operator is a map from R^n to itself. When viewed over the boolean semifiled, a valuated closure operator is equivalent to the closure operator of a matroid in the classical sense. We show that, over the tropical semifield, a closure operator is determined on the “maximal torus” $R^n$ (the set where no coordinates are equal to infinity). Moreover, a valuated closure operator over the tropical semifield offers a geometric interpretation as a nearest-point projection based on the tropical norm, and further provides an interpretation of tropical ideals as a form of valuated closure operator. These connections offer insight into applications in phylogenetics as well as the structure of bend congruences of matroids. Finally we further deduce a cryptomorphism for valuated matroids over the tropical semifield in terms of flats via the image of the closure operator.
Erin Dawson - Colorado State University
Title: Tropical Tevelev degrees
Abstract: Tropical Hurwitz spaces parameterize genus g, degree d covers of the tropical line with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product M_{0,n} cross M_{g,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. In this talk we will explore a method to perform this calculation of Tevelev degrees using the moduli spaces of tropical Hurwitz covers.
Danai Deligeorgaki - KTH Stockholm
Title: Ehrhart Positivity of Panhandle Matroids and a Bound for Paving Matroids
Abstract: Panhandle matroids are a specific lattice-path matroid corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. In recent work with Daniel McGinnis and Andrés R Vindas-Meléndez, we prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Mel\'endez, and Yin (2023). Our proofs rely solely on combinatorial techniques which involve determining intricate interpretations of certain set partitions. With similar techniques, we verify a conjecture of Ferroni (2022), which asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid, for the class of paving matroids.
Sergio Alejandro Fernandez de Soto Guerrero - TU Graz
Title: MathMagic: A Positroidal Action under a deck of cards
Abstract: Positroids are a subclass of Matorids that were born in the study of the non-negative Grassmanian by Postnikov in 2006, since then, there have been a plethora of combinatoric objects that index positroids, like Decored and Bicolored permutations, two generalizations of the symmetric group. Which ones can be studied through a bijection to a generalized permutahedron called Stelohedron, end up in useful ways to describe group action over a deck of cards, in this context, we can describe a different definition of invariants under a group action that allows us to explore its application with unusual ways of shuffling cards to develop new magic tricks. We will see how to define the generalization of the permutations and how to use it in magic!
Daniel Green-Tripp - University of Bristol
Title: Computing realisation numbers using tropical intersection theory
Abstract: The d-realisation number of a graph G counts, roughly speaking, the number of equivalent d-realisations of a generic d-realisation of G. It is known that this number is finite if and only if G is d-rigid. For a minimally 2-rigid graph, we give a way of computing this number as the tropical intersection number of the Bergman fan of the graphic matroid of G with its reciprocal.
Leo Jiang - University of Toronto
Title: Real matroid Schubert varieties, zonotopes, and virtual Weyl groups
Abstract: Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. We show that this variety has a surprisingly simple polyhedral model when the representation is over the real numbers. When the matroid representation comes from a root system, we identify the fundamental group of the variety with “virtual” analogues of the corresponding Weyl group. This is work in progress, joint with Yu Li.
Arne Kuhrs and Kevin Kühn - Goethe University Frankfurt
Title: Buildings, tropical linear spaces and valuated matroids - with a view towards a real version
Abstract: 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 (as the embedding and the dimension of the ambient projective space vary). 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 a recent follow-up work about a real analogue of the story. 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. This gives an interpretation of the signed Goldman-Iwahori space as the real tropical linear space associated to the universal realizable oriented valuated matroid.
Sofía Garzon Mora - FU Berlin
Title: On the spectra of Fine polyhedral adjoints
Abstract: Originally introduced by Fine and Reid, the Fine interior of a
lattice polytope got recently into the focus of research. Based on the
Fine interior, we study a modification of so-called adjoint polytopes and
define the Fine adjoint polytope of a polytope P as consisting of the
points in P that have lattice distance at least s to all its valid
inequalities. In this manner, we obtain a Fine Polyhedral Adjunction
Theory that is better behaved than its original analogue. Many existing
results in Polyhedral Adjunction Theory carry over. In this talk, we will
focus mainly on one of our conclusions obtained with simpler, more natural
proofs as is the case of the finiteness of the Fine spectrum. Namely, we
introduce the Q-codegree as an invariant arising in the context of toric
geometry, and discuss why it can take only finitely many values for
polytopes under certain conditions.
Anastasia Nathanson - University of Minnesota
Title: Permutation action on chow rings of matroids
Abstract: This talk will discuss the induced group action on the Chow ring of the matroid when given a matroid and a group of its matroid automorphisms. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.
Felix Röhrle - University of Tübingen
Title: A matroidal perspective on tropical Prym varieties.
Abstract: Following ideas from algebraic geometry, one can associate a tropical Jacobian variety to any tropical curve. As was outlined by Brannetti--Melo--Viviani, this construction can in fact be understood purely from a matroid theoretic perspective. In joint work with Dmitry Zakharov, we describe the analogous story for tropical Pryms varieties, thus generalizing the Jacobian version. The central idea is a modern view on signed graphic matroids which were originally introduced by Zaslavsky.
Karin Schaller - FU Berlin
Title: Nobodies are perfect, semigroups are not.
Abstract: NObodies are asymptotic limits of certain valuation semigroups. Their
construction depends on a given flag of subvarieties. We investigate toric
surfaces together with non-toric flags and determine when the associated
valuation semigroups are finitely generated. This is a joint work with K.
Altmann, C. Haase, A. Küronya, and L. Walter.
Lorenzo Vecchi - KTH Stockholm
Title: Equivariant polynomials in matroid theory
Abstract: Polynomial invariants in matroid theory can be upgraded to graded virtual representations by introducing a group of symmetries acting on the matroid polytope. While this seems to make the object you work with harder to handle, it turns out that it sometimes helps in computing the actual coefficients of your polynomials or in giving cleaner answers to questions regarding their non-negativity.
In the matroid landscape, starting from the characteristic polynomial, one can define the so-called Kazhdan-Lusztig-Stanley functions, which all admit an equivariant version. In this talk we are going to present concrete formulas on how to compute these objects.
Julian Weigert - University of Konstanz
Title: Equivariant Tutte Polynomial
Abstract: The usual Tutte polynomial T_M of a matroid M on {0,…,n} can be recovered in a geometrical way by considering certain classes in the cohomology ring of the permutohedral variety of dimension n. One can upgrade this formula to torus-equivariant cohomology and obtain an equivariant version of the Tutte polynomial. In this talk we will review how to assign equivariant cohomology classes to a matroid, following the work of Berget, Eur, Spink and Tseng from their article “Tautological Classes of Matroids”. We will then discuss their equivariant pushforward to a product of two projective spaces which can be seen as a way of intersecting the matroid classes with two special kinds of hyperplane pullbacks. In analogy to the non-equivariant setting, we define the equivariant Tutte polynomial of M to be a certain substitution of the resulting cohomology class. If time permits we will also look at some of the combinatorial properties of this polynomial such as for example a universality result with respect to deletion and contraction of elements of M.
This project is joint work with Mario Bauer, Matěj Doležálek, Magdaléna Mišinová, Semen Słobodianiuk, preprint: arXiv:2312.00913.