"Rational numbers ride the wave" by Patrik Nabelek of Oregon State University

Abstract: This talk will discuss the application of Abelian differentials of the first kind (i.e., holomorphic one form) on both compact Riemann surfaces of finite genus and infinite type Riemann surfaces with singularities to problems in coastal and ocean engineering, and physical oceanography. I will start by discussing how each Abelian differential on a compact Riemann surface determines a translation structure on the compact Riemann surface because the Abelian differentials on the compact Riemann surface generate flat cone metrics on the surface. The theory of Abelian differentials of the first kind, as well as Abelian differential of the second kind (a type of meromorphic differential on a compact Riemann surface), lead to a family of exact periodic and quasiperiodic solutions to the Korteweg—de Vries (KdV) equation and other completely integrable systems. I will also talk about a phenomenon called dispersive quantization which provides an interesting connection between nonlinear wave equations and rational numbers and periodic solutions to the KdV equation. I will discuss recent numerical and analytical techniques based on algebraic geometry, Riemann—Hilbert problems, and singular integral equations to the analysis of solutions to the KdV equation and other completely integrable systems. I will also discuss examples of ergodic theorems in the context of geodesics on translation surfaces, linear waves with random initial conditions, and solutions the Schrödinger equation. This talk will be based on joint work with Deniz Bilman, Tom Trogdon, Ken McLaughlin, Alexis Arlen, Tanner Fromcke, and Brandon Young.