Previous Special Year Seminar

For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $CP^2$ are collinear if and only if the sum of indices is divisible by $n$. We prove that for $n$ at...

The long-standing F-Conjecture asserts that there is a very simple description for the closed cone of effective curves on the moduli space M_{g,n}\bar of stable n-pointed curves of genus g as being determined by a finite collection of so-called F...

I will discuss various applications of a combinatorial model for the (torus equivariant) quantum K-theory of flag manifolds G/B, called the quantum alcove model. This is a uniform model for all Lie types, based on Weyl group combinatorics. It first...

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...