DE Seminar: Justin Webster (UMBC)
reporting on recent theory of infinite dimensional resonance
Title: Range Failure, Resolvent Growth, and The Story of Resonance in Infinite Dimensions
Abstract: Resonance is one of the oldest ideas in applied mathematics: drive an oscillator at the wrong frequency, and something breaks. But what should resonance mean in an infinite-dimensional system...how do we even define it?
This board talk will start from forced scalar ODEs and build toward very recent work on periodic PDE problems represented by semigroups on Hilbert spaces. We will discuss a taxonomy of no fewer than FOUR distinct resonance mechanisms!
The main surprise is that intrinsic damping/dissipation does not prevent resonance. Rather, the strength of dissipation determines how difficult resonance is to produce---and, in weakly dissipative systems---how ``smooth" the periodic force needs to keep things bounded.
Along the way, familiar functional analysis characters will appear in unexpected places: resolvent operators, semigroup stability, observability estimates, the Closed Range Theorem, Yosida’s Mean Ergodic Theorem, and a few other entertaining oddities. Come one, come all...nothing beyond MATH 225 is assumed. (Hold me to that!)
This is joint work with Boris Muha of the University of Zagreb (Hrvatska).
Related Publications:
Galdi, G.P., Muha, B. and Webster, J.T., 2026. From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems. arXiv preprint arXiv:2605.12892.
Mosny, S., Muha, B., Schwarzacher, S. and Webster, J.T., 2024. Time-periodic solutions for hyperbolic-parabolic systems. Accepted May 2026: J. European Mathematical Society. arXiv preprint arXiv:2412.18801.
Benson, I. and Webster, J.T., 2024. Resonance and periodic solutions for harmonic oscillators with general forcing. Accepted December 2025: J. European Mathematical Society. arXiv preprint arXiv:2407.17144.