Identifying engineering (im)possibilities with geometry and topology

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

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.

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.

Control-Lyapunov or barrier functions yield checkable sufficient conditions for stability or safety to be achievable with feedback control, but not checkable necessary conditions. I will describe new checkable necessary conditions for achievability of stability or safety, along with ongoing work on sufficient conditions for periodic orbit stabilizability.

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

Bio: Matthew D. Kvalheim is an assistant professor in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County. He previously held postdoctoral positions in the Departments of Mathematics at the University of Michigan and in the GRASP Laboratory at the University of Pennsylvania. He received an MS in Mathematics and MS and PhD degrees in Electrical and Computer Engineering from the University of Michigan. He is broadly interested in dynamical systems, control theory, and applied topology and geometry.