Fermi Surfaces and Infinite Genus Riemann Surfaces

Joel Feldman

Abstract: Riemann's analysis of finite genus one dimensional complex manifolds is a mathematical gem. These lectures will be an introduction to a class of infinite genus Riemann surfaces to which much of the classical theory extends. I will concentrate on a family of examples that arise in the study of the spectrum of Schrödinger operators, which are differential operators that play a central role in modeling crystals.

I am hoping that most of the lectures will be accessible to graduate students. I will cover as much of the following outline as time permits.

Periodic Schrödinger Operators
- brief physical motivation for studying periodic Schrödinger Operators
- the idea of the Bloch decomposition
- outline of how to make the Bloch decomposition rigorous
- a nontrivial example - the Lamé Equation
Fermi Surfaces for Space Dimension d=2
- the free Fermi surface as a one complex dimensional curve in two complex dimensions
- the effect of turning on interactions.
Review of Finite Genus Riemann Surfaces
Infinite Genus Riemann Surfaces
- a set of axioms
- Theta functions
- Zeros of the Theta Function and Riemann's Vanishing Theorem
- the Torelli Theorem