Spring 2025 Colliquia
April 11, 2025
Amol Aggarwal, Clay Mathematics Institute and Columbia University
Topic: The Toda Lattice as a Soliton Gas
Time: 4:00-5:00 pm
Location: LL08 Barnum Complex
Reception: JCC501
Abstract: A basic tenet of integrable systems is that, under sufficientl yirregular initial data, they can be thought of as dense collections of many solitons, or “soliton gases.” In this talk we focus on the Toda lattice, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system, under certain random initial data, can be interpreted through solitons, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.
March 7, 2025
Jen Hom, Georgia Institute of Technology
Topic: 3-manifolds, groups, and Heegaard Floer homology
Time: 4:00-5:00 pm
Location: JCC 270
Reception: JCC 501
Abstract: We will consider various ways to build 3-manifolds. Under the operation of connected sum, the set of 3-manifolds forms a monoid, and modulo an appropriate equivalence relation, this monoid becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.
February 7, 2025
Nick Trefethen, Harvard University
Topic: Rational Approximation and the AAA Algorithm
Time: 4:00-5:00 pm
Location: JCC 270
Reception: JCC 501
Abstract: Approximation by rational functions used to be mainly a theoretical subject, but with the introduction of the AAA algorithm in 2018, it became computationally practical and indeed easy. The implications for what we can do numerically are enormous. This talk will outline the algorithm and demonstrate its application to a collection of problems. We can also use it to demonstrate the potential theory that underlies the theory of rational approximation, a topic that goes back to Joseph Walsh at Harvard nearly a century ago.