Previous Special Year Seminar

I will show that any Schubert or Richardson variety R in a flag manifold G/P is equivariantly rigid and convex. Equivariantly rigid means that R is uniquely determined by its equivariant cohomology class, and convex means that R contains any torus...

We describe three constructions of spherical roots of spherical varieties via embddings, Borel orbits, and harmonic analysis, and give a hint for why they yield the same results. Then we describe more recent extensions of this theory to arbitrary G...

In part 1, I will survey the history of total positivity, beginning in the 1930's with the introduction of totally positive matrices, which turn out to have surprising linear-algebraic and combinatorial properties. I will discuss some modern...

In this talk I will discuss various connections between the dynamics of integrable Hamiltonian flows, gradient flows, and combinatorial geometry. A key system is the Toda lattice  which describes the dynamics of interacting particles on the line. I...

Configuration spaces of points in graphs are nonsmooth analogs of braid arrangements, appearing in robotics applications and in theory of moduli spaces of tropical curves. While their cohomology is extremely difficult to understand, and depends on...

Suppose given a class of finite combinatorial structures, such as graphs or total orders. Nate Harman and I recently introduced a notion of measure in this context: this is a rule assigning a number to each structure such that some axioms are...

The Kadomtsev-Petviashvili (KP) Equation has deep connections to algebraic curves, with solutions constructed from Riemann theta functions in the style of Krichever. As a curve undergoes tropical degeneration, its theta function simplifies to a...

Given a triple (X,π,s) consisting of a homogeneous space X=G/P, a dominant weight π giving a projective embedding of X, and a reduced expression s for the minimal coset representative of w_0 in the parabolic quotient W/W_P, we construct a polytope...

We will discuss combinatorial and algebraic aspects of regular Hessenberg varieties, a large class of subvarieties of the flag variety G/B. For the special case of Peterson varieties, we show their equivariant structure constants are non-negative...

A real plane algebraic curve C is called expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C. We give a necessary and sufficient criterion for expressivity (subject...