Fall Quarter 2001

Tuesdays and Thursdays, 9:00 - 10:20

Chapters 0 and 1 with References in .pdf (Completed November 1, 2001, incorporating a kind input from Sunny Fawcett)

Chapter 2 in .pdf (Last section still incomplete, as of November 1, 2001)

Moduli Theory Lecture Notes (whole) in .pdf (whatever written by November 1, 2001)

Text: None, but to be published in the future.

Office Hours: Whenever you find me in my office.

Course Description: The course is planned as an introduction to the theory of Riemann surfaces and their moduli. Instead of taking the historical approach, we will jump into the late 20th century and examine the new approach to the moduli theory developed by Thurston, Witten, Kontsevich and Grothendieck.

The goal is to give a combinatorial description of the moduli spaces of Riemann surfaces.

The course is not intended as an introduction to the language of algebraic geometry such as schemes, coherent sheaves, and their cohomologies. Those who are interested in learning them will be unfortunately disappointed. (However, I'm happy to run a reading course MAT 299 if you need to learn "le langage des schemas.")

As the list of topics below shows, the course will present the integration of the ideas from analysis, number theory, geometry, combinatorics, topology, and mathematical physics.

Prerequisite: Knowledge from the undergraduate mathematics courses such as MAT 127ABC, 145, 147, 150ABC, 167, and 185AB (or their equivalent) is assumed. MAT 201ABC (or 203ABC), 205, and 250ABC would be great if you have taken them, but they are not required. I will try to make the course accessible to the first-year students (without any guarantee, though).

Topics discussed: Riemann surfaces, moduli of Riemann surfaces, the Riemann zeta function and its special values, the gamma and beta functions, the Selberg integral formulas, Aomoto integrals, GUE random matrices, Feynman diagram expansion of a Quantum Field Theory, string theory, geometric Galois theory, algebraic curves defined over the field of algebraic numbers, and graph theory. (Please note: I did not say "topics covered"!!!)

Course grade: Problems will be given in class, including the open problems I'm working now. If you solve one of the open problems, then I'll recommend to the Department that you receive an instantaneous Ph.D. The course grade will be based on your results, attempts, and/or attendance.