Geometry and Integrability of Hamiltonian and Gradient Flows

This special year seminar at the Institute for Advanced Study features a lecture by Anthony Bloch from the University of Michigan exploring the connections between integrable Hamiltonian flows, gradient flows, and combinatorial geometry. Delve into the Toda lattice system, which describes the dynamics of interacting particles on a line, and discover how it can be viewed as gradient flows related to convex polytope geometry and total positivity theory. Learn how this mathematical framework, originating in linear algebra and matrix theory, extends to representation theory with applications in combinatorics, small vibrations, and high energy physics. The presentation demonstrates how these dynamics can prove significant results in total positivity theory on flag manifolds, drawing from Bloch's recent collaboration with Steven Karp and earlier work with Brockett, Flaschka, and Ratiu. The seminar is scheduled for April 23, 2025, at 2:00pm in Simonyi 101.