Short course on Compact Riemann surfaces and the KdV equation
by Dr. Ian McIntosh (University of York, UK)
Dr. Ian McIntosh gave a series of 8 lectures on the theory of compact Riemann surfaces, ending with applications to the KdV equation. The lectures were intended as an introduction to this theory, which has its roots in classical mathematics of algebraic curves and complex function theory in the 19th century. It is now at the forefront of research in geometry and those nonlinear partial differential equations known as integrable systems.
The lectures began with material accessible to graduate students from any scientific field, namely the basic definitions of Riemann surfaces and their meromorphic functions and differential forms. The residue theorem and period integrals followed, leading to the definition of the Jacobi variety and the Abel map. A proof of Abel’s Theorem was given in detail. The Riemann theta function was defined and some of its properties derived. Finally, the famous theta functions of the KdV equation were constructed.
A remarkable amount of material was covered, ranging from basic definitions to concepts which are still the focus of research. Detailed lecture notes were provided. Discussions of exercises (proposed by the lecturer) and reference material were also very much appreciated by the audience of around 20 participants.