Dimitris Giannakis (Dartmouth): CAM and DSI Joint Colloquium

Please join us for CAM and DSI joint colloquium.

Friday, April 25
12:00pm – 1:00pm
Jones 303
5747 S. Ellis Avenue

Abstract: A key structural property of Koopman (composition) operators of dynamical systems is that they are multiplicative on spaces of observables with a product structure. This property implies, for instance, that the point spectrum is an abelian group and that the corresponding eigenfunctions also form a group under pointwise function multiplication. Computational approximation techniques for Koopman operators typically do not preserve multiplicativity in a strict sense, yet one might expect that for approximations that are sufficiently “close” to the original operator products of eigenvalues and eigenfunctions should be useful for building models for the evolution of observables. Using this as a working hypothesis, in this talk we present a scheme for approximating the Koopman evolution of observables that is based on a lift to a Fock space where regularized Koopman operators are multiplicative with respect to the tensor product. This Fock space is generated by a reproducing kernel Hilbert space of observables endowed with the structure of a coalgebra with respect to the tensor product, and the structure of a Banach algebra with respect to the pointwise product of functions. The resulting approximation scheme can be cast in the form of a tree tensor network allowing for efficient computation in high-dimensional spaces generated multiplicatively from a modest collection of approximate Koopman eigenfunctions. We illustrate this approach with applications to measure-preserving ergodic flows on tori.

Bio: My primary research areas are operator methods for dynamical systems and mathematical data science. I joined Dartmouth College in September 2021 as Jack Byrne Professor in Mathematics. Previously, I was an Associate Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University. As of July 2024, I am serving as Chair of Dartmouth’s Department of Mathematics.