Integrable Systems

Lecture 1: Riemann-Hurwitz Formula

Description: We introduce Riemann surfaces and covering surfaces, focusing on the example of hyperelliptic surfaces that are obtained from the zero set of a polynomial on two variables. We defined branching points for covering surfaces. Then, we present the Riemann-Hurwitz formula through the Euler characteristic.

Lecture 2: Torus KdV Solution

Description: We build a solution of the KdV equation based on the Riemann theta functions. The terms that enter the theta function come from objects defined on a Riemann surface. In particular, these terms are Abelian integrals and we focus on the genus one case. In the next lecture, we will show that our construction is indeed a solution of the KdV equation.

Lecture 3: The Baker-Akhiezer function

Description: We define the Baker-Akhizer function for Riemann surfaces, focusing on the genus one case. This function is completely characterized by the location of the poles, the location of the essential singularities, and the behaviour of the function near the essential singularities. We use the BA function to solve the KP equation. In particular, we first show that the KP equation is a consistency condition for a pair of linear differential operators. Then, we show that the BA function is a simultaneous solution of these pair of differential operators. Finally, from the BA function, we are able to extract an expression for the solution of the KP equation in terms the Riemann theta function.

Lecture 4: Riemann-Roch theorem

Description: We show that the Baker-Akhiezer (BA) function is uniquely determined by the prescribed singularity information up to a multiplicative constant. This result is powered by the Riemann-Roch theorem. We begin the talk by recalling some fact about meromorphic functions and differentia forms for Riemann surfaces. Then, we give an elementary argument for the Riemann-Roch theorem. We finish by proving the original statement for the BA functions.