In this talk we will begin by defining elliptic curves from three perspectives. The first is as the space of solutions to certain polynomial equations in two complex variables, the second is as a parallelogram with certain sides identified by translation, and the third is as the complex numbers modulo some period lattice. There are interesting classes of functions on these surfaces called elliptic functions.
We will then discuss some problems in physics that can be solved using elliptic functions. In particular, we will discuss how elliptic functions appear in the solutions to nonlinear oscillators, the pendulum, the Euler top equation, as well as problems in wave prorogation, and the geodesic equation for objects orbiting a spherically symmetric body according to general relativity.
We will then discuss how the Kovalevskaya top equation necessitated the use of objects called hyper elliptic curves and Riemann surfaces to solve nonlinear equations in a similar. Sofya Kovalevskaya’s work later inspired Boris Dubrovin an approach to a nonlinear PDEs called completely integrable systems based on transforming them into families of finite dimensional ODE systems.
We will then discuss an object called a moduli spaces, what his has to do with a peculiar water wave called an undular bore, and discuss what this all has to do with eating donuts.
(Bring a donut if you want to participate by eating one, preferably with two holes!)