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...
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 oriented...
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...
