Fall 2023 GRASP SFI: Matthew D. Kvalheim, University of Maryland, Baltimore County, “Discovering engineering (im)possibilities with geometry and topology”

This was a hybrid event with in-person attendance in Levine 307 and virtual attendance…

ABSTRACT

I will describe engineering (im)possibilities discovered with geometry or topology. These provide or revoke “hunting licenses” for the search of quantities of interest in three contexts: feedback control, applied Koopmanism, and deep neural network autoencoders.

Control-Lyapunov or barrier functions yield sufficient conditions for stability or safety to be achievable with feedback control, but cannot determine if this is not achievable. I will present user-friendly “tests” to determine this, along with ongoing work on sufficient conditions for periodic orbit stabilizability.

An open problem for Koopman methods has been to determine the class of dynamical systems that are globally linearizable in the sense of admitting an embedding into a linear system on a Euclidean space. I will present a solution for the case of linearizing compact invariant sets or attractor basins.

Topological obstructions dictate that autoencoders cannot provide nonlinear dimensionality reductions with small errors, and yet, the wide practical applicability of the method evidences remarkable empirical success. I will offer a resolution to this apparent paradox.

This is joint work with P. Arathoon, A. M. Bloch, D. E. Koditschek, and E. D. Sontag.