The Department of Applied Mathematics weekly seminar is given by scholars and researchers working in applied mathematics, broadly interpreted.
Title: Integrable Hamiltonian and gradient flows and total positivity
Abstract: In this talk I will discuss various connections between the dynamics of integrable (solvable) Hamiltonian flows, gradient flows, and geometry. A key example will be the Toda lattice flow which describes the dynamics of interacting particles on the line. I will show how versions of this can also be viewed as gradient flows and relate the flow to the geometry of convex polytopes as well as to the theory of total positivity. The latter theory has its origins in linear algebra and matrices, all of whose minors are positive. This simple idea has fascinating generalizations to representation theory and applications in combinatorics, small vibrations and high energy physics. The type of dynamics discussed here turns out to be able to prove interesting results in the general theory of total positivity. I will also discuss links to general dissipative dynamics and to deep learning.