Geometry and Integrability of Hamiltonian and Gradient Flows - Anthony Bloch

Special Year Seminar I 2:00pm|Simonyi 101 Topic: Geometry and Integrability of Hamiltonian and Gradient Flows Speaker: Anthony Bloch Affiliation: University of Michigan Date: April 23, 2025 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 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 the theory of matrices all of whose minors are positive. This 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 on flag manifolds. This talk is based on recent work with Steven Karp and earlier work with Brockett, Flaschka and Ratiu.