Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, March 8, 2021 - 12:00 to 12:50

Location:

Zoom. Please contact Christine Escher for a link.

Local Speaker:

Abstract:

Are donuts with multiple holes and nonlinear waves related? It turns out the answer is yes!

The KdV equation is an important non-linear partial differential equation with applications in mathematical physics that was initially derived to describe the behavior of peculiar waves called solitons. It has a class of solutions called "finite gap solutions" that are determined by compact Riemann surfaces, which can be thought of as compact surfaces with a finite number of holes. In this talk I will present an explicit construction of a family of genus 2 Riemann surfaces, discuss how some aspects of the general theory of Riemann surfaces manifest in this context, and use these surfaces to construct examples of "two gap" solutions to the KdV equation. I will also show some numerical calculations to illustrate how these solutions behave.