Bernoulli Lecture – Internal waves and homeomorphism of the circle

25 March 2021

Part of the Semester : Dynamics, Transfer Operators, and Spectra

17:15 – 18:15


Maciej Zworski, University of California, Berkeley


The connection between the formation of internal waves in fluids and the dynamics of homeomorphisms of the circle was investigated by oceanographers in the 90s and resulted in novel experimental observations (Maas et al, 1997). The specific homeomorphism is given by a

chess billiard” and has been considered by many authors (John 1941, Arnold 1957, Ralston 1973, … , Nogueira-Troubetzkoy 2020). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!) and, using a model of tori and of zeroth order pseudodifferential operators, it has been a subject of recent research, first by Colin de Verdière-Saint Raymond 2020 and then by Dyatlov, Galkowski, Wang and the speaker. In these works, many facets of the relationship between hyperbolic sources and sinks for the classical dynamics and internal waves in fluids were explained. I will present some of these results as well as some numerical discoveries (including those of Almonacid-Nigam 2020). I will also describe various open problems.