Special Year 2024-25: Algebraic and Geometric Combinatorics

Abstract: The derived category of a variety is an important and difficult invariant. In this talk, I discuss a purely convex-geometric and combinatorial approach to these categories for toric varieties. Along the way, we will run into the curious...

Abstract: Delta matroids are a generalization of matroids, encompassing combinatorial structures such as matchings of graphs and principal minors of symmetric matrices. In this talk, I will discuss a notion of valuated Delta matroids and their...

Abstract: Baker and Bowler (2019) showed that the Grassmannian can be defined over a tract, a field-like structure generalizing both partial fields and hyperfields.

This notion unifies theories for matroids over partial fields, valuated matroids, and...

Abstract: In this talk, I will first review previous works on matroid representations initiated by Tutte in 1958, generalized by Dress and Wenzel in the 1980s, and refined by Baker, Bowler, and Lorscheid more recently using the language of F1...

Abstract: I will share with you a connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise from topological graph theory. The main...

Abstract: I will introduce the external activity complex of an ordered pair of matroids on the same ground set. This is the combinatorial analogue of the Schubert variety of an ordered pair of linear subspaces of a fixed space, which is also new...

Abstract: The groundbreaking results by Huh (further extended in joint work with Adiprasito and Katz) allowed to associate to a matroid a class in the Chow ring of the permutohedral variety. The technique turned out to be especially powerful, as...

Abstract: We study the tropicalization of principal minors of positive definite matrices over a real valued field. This tropicalization forms a subset of M-concave functions on the discrete n-dimensional cube. We show that it coincides with a linear...

Abstract: 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...

Abstract: This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety. A motivating example are linear embeddings of complements of hyperplane arrangements. 

We can prove that the tropicalisation...