Georgia Institute of Technology College of Sciences

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the Korteweg-de Vries equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

We study random configurations of N soliton solutions q_N(x,t) of the KdV equation. The randomness appears in the scattering (linear) problem, which is used to solve the PDE: the complex eigenvalues are chosen to be (1) i.i.d. random variables sampled from a probability distribution with compact support on the complex plane, or (2) sampled from a random matrix law. 

Next, we consider the scattering problem for the expectation of the random measure associated to the spectral data, in the limit as N -> + infinity. The corresponding solution q(x,t) of the KdV equation is a soliton gas. 

We are then able to prove a Law of Large Numbers and a Central Limit Theorem for the differences q_N(x,t)-q(x,t).

This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).

The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial. More precisely, we will show how Birman, Wenzl, and Murakami constructed the BMW algebra and a trace function that yields the Kauffman polynomial. We will discuss the significance of the Kauffman polynomial as well as some relationships between the BMW, Hecke, and Jones algebras.

The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds, and the existence of non-trivial homomorphisms $\pi_1(M)\to SU(2)$ is a great way of measuring the non-triviality of a three-manifold $M$. It is known that if an integer homology 3-sphere is either Seifert fibered or toroidal, then irreducible representations do exist. In contrast, the existence of SU(2)-representations for hyperbolic homology spheres has not been completely established. With this as motivation, I will talk about partial progress made in the case of hyperbolic homology spheres realized as branched covers. This is joint work with Sudipta Ghosh and Zhenkun Li.

The flag variety G/B plays an outsized role in representation theory, combinatorics, geometry and algebra. Hessenberg varieties form a special class of subvarietes of the flag variety, arising in diverse contexts. The cohomology ring of a semisimple Hessenberg variety is recognized to be a representation of an associated finite group, and is related to the expansion of some special polynomials in terms of other well-known polynomial bases. These varieties may have pathological behavior, and their basic properties have been characterized only in restricted cases. Matrix Hessenberg schemes in type A consist of a lift of these varieties to G = Gl(n, C), where we can use the coordinate ring of matrices to study them.

In this talk, we present a full characterization of matrix Hessenberg schemes over the minimal sheet of Lie(G) in type A. We show that each semisimple matrix Hessenberg scheme lies in a flat family with a nilpotent matrix Hessenberg scheme, which in turn allows us to study their geometric properties. We describe the schemes fully in terms of Schubert varieties and opposite Schubert varieties, both well-known subvarieties of G/B. More subtly we characterize combinatorially which matrix Hessenberg schemes are reduced. These results are joint with Martha Precup at Washington University, St. Louis.