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"...
We analyze a model of qubits which we argue has an emergent
quantum gravitational description similar to the fermionic
Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as
the quantum $q$-spin model because it features $q$-local...
Associated to a star-shaped domain
in ℝ2nR2n are two increasing sequences of
capacities: the Ekeland-Hofer capacities and the so-called
Gutt-Hutchings capacities. I shall recall both constructions and
then present the main theorem that they are the...
In the early 2000's, Bertolini and Darmon introduced a new
technique to bound Selmer groups of elliptic curves via level
raising congruences. This was the first example of what is now
termed a "bipartite Euler system", and over the last decade
we...
