Geometry without Geometry - Titles and abstracts

Omid Amini (École Polytechnique): Combinatorial Newton-Okounkov bodies

Newton-Okounkov bodies were introduced by Kaveh-Khovanskii and by Lazarsfeld-Mustata in order to describe asymptotic geometry of a given algebraic variety by encoding in a convex body the asymptotic behavior of its embeddings. Tropical geometry on the other hand aims at describing large scale asymptotics of families of algebraic varieties. The tropical limit remembers essential information about the geometry close to infinity of the individual members of the family. We develop a purely polyhedral geometric formulation of higher rank valuations which underline the definition of Newton-Okounkov bodies. Using the set-up, we define valuations on tropical varieties and attach Newton-Okounkov bodies to them. I will present the framework and discuss connections with combinatorial Hodge theory and with higher dimensional version of divisor theory on graphs.

Based on joint works with Hernan Iriarte and Matthieu Piquerez.

Marie-Charlotte Brandenburg (KTH Stockholm): Quotients of M-convex sets and M-convex functions

The combinatorial structure of a subspace arrangement can be captured by a polymatroid. The polymatroid arising from the image of the subspace arrangement under a linear map is in an intricate relation with the original polymatroid. This relation leads to the notion of quotients for submodular functions and M-convex sets, generalizing matroid quotients and flag matroids.

In this talk, we will discuss several equivalent characterizations of quotients of M-convex sets, and how these characterizations suggest multiple (potentially non-equivalent) candidates for the definition of quotients of M-convex functions and valuated matroids.

This talk is based on joint work with Georg Loho and Ben Smith.

Giulia Codenotti (FU Berlin): Lattice reduced polytopes and the flatness question

We call a convex body lattice reduced if it does not strictly contain any other convex body of the same lattice width--a parameter studying "how flat" a convex body is with respect to the lattice of integer points. We show structural properties of lattice reduced convex bodies: that they are polytopes, with not too many vertices; and then investigate the connection to the flatness problem: determining, for each dimension, the maximum lattice width of convex bodies without interior integer points. We will then delve into further directions of study and open questions.

Alex Fink (Queen Mary University): Speyer's g conjecture and Betti numbers for a pair of matroids

In 2009, looking to bound the face vectors of tropical linear spaces, Speyer introduced the g-invariant of a matroid. He proved its coefficients nonnegative for matroids representable in characteristic zero and conjectured this in general. Later, Shaw and Speyer and I reduced the question to positivity of the top coefficient. This talk will overview work being written up with Berget that proves the conjecture.

Geometrically, the main ingredient is a variety obtained from projection away from the base of the matroid tautological vector bundles of Berget--Eur--Spink--Tseng, and initial degenerations thereof. Combinatorially, it is an extension of the definition of external activity to a pair of matroids and a way to compute it using the fan displacement rule on tropical Chern classes. The work of Ardila and Boocher on the closure of a linear space in (P^1)^n is a special case.

Mario Kummer (TU Dresden): Spaces of Lorentzian polynomials

We characterize the topology of the space of Lorentzian polynomials with a given support in terms of the local Dressian. We prove that this space can be compactified to a closed Euclidean ball whose dimension is the rank of the Tutte group. Finally, we show that a compactification proposed by Brändén is in general not homeomorphic to a closed ball. This is part of a joint work with Matt Baker, June Huh and Oliver Lorscheid.

Dustin Ross (San Francisco State University): Ehrhart fans

Motivated both by lattice point counting in discrete geometry and by Euler characteristics of sheaves in algebraic geometry, we introduce a new class of fans that we call Ehrhart fans. The defining property of an Ehrhart fan is that it admits a canonical polynomial map from the group of piecewise linear functions on the fan to the integers. In this talk, we will introduce Ehrhart fans and discuss how they provide a setting in which Euler characteristics of matroids can be generalized and studied alongside Euler characteristics of smooth complete toric varieties. This is joint and ongoing work with Melody Chan, Emily Clader, and Carly Klivans.

Victoria Schleis (Durham University): Tropical Quiver Grassmannians

Grassmannians and flag varieties are important moduli spaces in algebraic geometry. Quiver Grassmannians are generalizations of these spaces arise in representation theory as the moduli spaces of quiver subrepresentations. These represent arrangements of vector subspaces satisfying linear relations provided by a directed graph.

Using tropical geometry, we introduce matroidal and tropical analoga of quivers and their Grassmannians obtained in joint work with Giulia Iezzi; and describe them as affine morphisms of valuated matroids and linear maps of tropical linear spaces. Further, I will discuss first steps towards a polyhedral characterization of morphisms, and give an overview of recent work in progress on applying and extending the theory.

Petra Schwer (Heidelberg): The impact of move-spaces on algebraic properties of affine Coxeter groups

Every Coxeter group W admits a natural action by affine transformations on some finite dimensional real vector space V. The move-space of an element w in W is the collection of all vectors v in V by which some point x in V is moved, that is all v s.th. there is an x such that w.x=x+v. I will explain in this talk how this space impacts the structure of reflection length and conjugacy classes in W.

Josephine Yu (Georgia Tech): Real Tropical Geometry, principal minors, and matroids

Tropicalization is a degeneration process used to examine the exponential behavior at infinity of algebraic and semialgebraic sets, defined by polynomial equations and inequalities. This process uncovers combinatorial structures within these geometric objects. When applied to the principal minors of positive definite matrices, tropicalization reveals a discrete convexity structure and makes connections to flag varieties and representability of (poly)matroids. It also helps in discovering new polynomial inequalities among the principal minors.