Video Lectures

I will discuss some regular subdivisions of the permutahedron, one for each Coxeter element in the symmetric group. These subdivisions are "Bruhat interval" subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat...

In his 2018 paper "Some Schubert shenanigans" Richard Stanley asked for the asymptotic behavior of the maximal principle specialization of a Schubert polynomial. Motivated by this, still open, question we explore the generalization to Grothendieck...

The general rule for the interactions between tropical geometry and moduli spaces of course is the following: everything you may wish is going to work like a charm in genus zero, and break down horribly in higher genus. This is the case for the...

The spectral norm on the group of Hamiltonian diffeomorphisms of a symplectic manifold is defined via a homological min-max process on the filtered Floer homology. Based on the spectral norm one defines the spectral capacity of domains which is...

We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with...

We present a new method to solve algorithmic and combinatorial problems by (1) reducing them to bounding the maximum, over x in {-1, 1}^n, of homogeneous degree-q multilinear polynomials, and then (2) bounding the maximum value attained by these...

For a smooth projective variety, the classical Hirzebruch--Riemann--Roch (HRR) theorem asserts the isomorphism between the Chow ring and the Grothendieck K-ring of vector bundles over rational coefficients. For certain toric varieties, we show an...

There has been a lot of recent work connecting the two pipe dream formulae for Schubert polynomials (classic and bumpless). One strand is the hybrid pipe dreams of Udell and myself, giving 2n

different pipe dream formulae; an orthogonal one is the...

Chow rings of toric varieties feature a rich combinatorial structure of independent interest. We will begin by surveying four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We will illustrate...

The recent breakthrough of Limaye, Srinivasan and Tavenas (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this...