Mathematical Sciences Dept. Colloquium - Paul Milewski, Penn State University
11:00 a.m. to 12:00 p.m.
Mathematical Sciences Colloquium
Speaker: Paul Milewski, Penn State
Friday, May 2, 2025
11:00am - 12:00pm
Stratton Hall 202
Title: Resonance of surface water waves in cylindrical containers
Abstract: Waves sloshing in a container of rectangular cross-section can behave very differently than those in a circular one, beyond the linear modes of the system. This talk covers one aspect of how geometry affects the nonlinear evolution of waves. Nonlinear resonance is a mechanism by which energy is continuously exchanged between a small number of wave modes and is common to many nonlinear dispersive wave systems. In the context of free-surface gravity waves such as ocean surface waves, nonlinear resonances have been studied extensively over the past 60-years, almost always on domains that are large compared to the characteristic wavelength (and were mentioned in the award of the 2021 Nobel Prize in Physics to Klaus Hasselmann). In this case, the dispersion relation dictates that only quartic (4-wave) resonances can occur. In contrast, nonlinear resonances in confined three-dimensional geometries have received relatively little attention, where, perhaps surprisingly, stronger 3-wave resonances do occur. We will present the results characterizing the configuration and dynamics of resonant triads in cylindrical basins of arbitrary cross sections, demonstrating that these triads are ubiquitous, with a rectangular cross section being an exception where they do not occur! Most of the talk will be accessible to a general audience with undergraduate mathematical knowledge.
Bio: Paul Milewski is a Professor and current Department Head of the Department of Mathematics at Penn State. He received his B.Sc. and M.Sc. in Aerospace Engineering at Boston University and his Ph.D. (1993) from Mathematics at M.I.T. Prior to joining Mathematics at Penn State in 2023, he had positions in the Departments of Mathematical Sciences at the University of Bath (UK, 2011-2023), and in Mathematics at Wisconsin-Madison (1995-2011) and Stanford (1993-1995). He has held visiting positions, among others at IMPA (Brazil) and ENS (France). He was recipient of a Royal Society Wolfson Fellowship and a Sloan Fellowship. His research is in applied and computational mathematics, mainly in nonlinear waves in fluids, but also with interests across mathematical modeling of physical and biological systems, and data science.