School of Mathematics

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-algebra. It...

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 result is...

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. These...

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 certain...

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 slice of...

In 1987, Hofer and Zehnder showed that for any smooth function H

on ℝ2n, almost every compact and regular level set contains at least one closed characteristic. I'll show that, when n=2, almost every compact and regular level set contains at least...

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...

I will discuss the Hanna Neumann conjecture of the 1950's and some tools in graph theory that I used to solve it.  The tools include sheaf theory on graphs, Galois theory for graphs, and the preservation of "local properties" under base change (for...

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 knows the...

The foundation of a matroid is an algebraic invariant that controls representations over any partial field, hyperfield, or more generally, any pasture. We show that, under certain conditions, the foundation of a generalized parallel connection of...