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...
After a gentle introduction to matroids, I will present parts of
a new OSCAR software module for matroids through several examples.
I will focus on computing the moduli space of a matroid which is
the space of all arrangements of hyperplanes with...
Computational problems exhibit a diverse range of behaviors in
terms of how quickly and effectively they can be solved. What
underlying mathematical structure (or lack thereof) in a
computational problem leads to an efficient algorithm for
solving...
A toric vector bundle is a torus equivariant vector bundle on a
toric variety.
We begin by recalling the classification of toric vector bundles
due to Klyachko. The Klyachko data of a toric vector bundle can be
interpreted as a "piecewise linear map"...
