MAT 280 - 01 Hurwitz Numbers and Moduli Theory

An Introduction

Fall Quarter 2009, CRN = 43417

Time: TR 10:30 - 11:50

MSB 2112

Instructor: Motohico Mulase, with Naizhen Zhang as Special Teaching Assistant

Office: MSB 3103

Phone: 752-6324

First Class Meeting: Tuesday, September 29, 2009. There will be no organizational meeting. We immediately start mathematics.

Instructor Office Hours: Thursdays 2:00PM - 5:00PM in MSB 3103.

Prerequisite: Graduate standing, or consent of the instructor. More specifically, there is no prerequisite in terms of high-level mathematical knowledge. Please come with a lot of curiosity.

Topics Covered (tentative):

Week 1: No prerequisite! So we start with reviewing the notion of topological covering of a punctured sphere, fundamental groups, the Euler characteristic of a surface, Riemann surfaces, and holomorphic and meromorphic functions in one variable. We then derive the Riemann-Hurwitz formula for the Euler characteristic of a Riemann surface. Everything is totally elementary.

Week 2: ?

Week 3: ??

Course Plan:

A 3,800-year old clay tablet called Plimpton 322 suggests that ancient Babylonians were fascinated by Pythagorean numbers, and that they must have known some mechanism to generate them.

In the 21st century, we are fascinated by topological invariants, and we are working to find mathematical mechanisms that generate them.

This course is intended to be a leisurely introduction to Gromov-Witten invariants and a new mechanism to generate some of these invariants.

The common difficulty for a student to learn a new exciting mathematics is the daunting amount of materials to be digested even for understanding the definition of the subject. Instead of taking a textbook style approach, we are planning to launch a course that serves as an excursion. We will focus ourselves to the study of Hurwitz numbers, and use our insight on these numbers as a guide to see what's going on in the front line.

Over the century old idea of Hurwitz is to study a concrete object for understanding an abstract notion. Hurwitz proposed to use Hurwitz spaces for the study of the moduli spaces of Riemann surfaces. The former is a concrete object easy to deal with, while the latter is a mysterious object even today. Hurwitz numbers count topological covers of a punctured two-dimensional sphere with prescribed combinatorial data. The amazing fact is that Hurwitz numbers are indeed quite non-trivial examples of Gromov-Witten invariants.

An undergraduate mathematics major student can easily understand Hurwitz numbers. Our excursion starts from here. The richness of the subject can be seen by its relation to topology of surfaces, combinatorics of partitions, representation of symmetric groups, integrable nonlinear PDEs such as KdV and KP equations, moduli theory of Riemann surfaces, Witten-Kontsevich theory, Gromov-Witten theory, Random Matrix Theory, and topological string theory.

Following the path of Hurwitz numbers in our excursion, we can see some of the high points of modern mathematics in a much easier way than originally presented. For example, we present a short and easy proof of the Witten Conjecture/Kontsevich theorem on the intersection numbers of the moduli spaces of algebraic curves following arXiv:0908.2267 math.AG. It has been also pointed out that some of the quantum knot invariants are obtained by similar techniques. Our goal is to climb up high and to see the beautiful scenery of these mathematics.

Remark for experts: Our true goal is to explore a new theory of Eynard-Orantin Topological Recursion. The Virasoro Conjecture has been a central theme of Gromov-Witten theory for the past 10+ years. This conjectural formula is trivially true for Calabi-Yau spaces, and carries absolutely no information about Gromov-Witten invariants of these spaces. Thus we need something else to generate the invariants. At least for toric Calabi-Yau 3-folds, there is a conjectural effective recursion that calculates the Gromov-Witten invariants. This new formula is unrelated to the known (trivial) Virasoro constraint condition for Calabi-Yau spaces. It is a direct generalization of the Witten-Kontsevich theory, and is originated in Random Matrix Theory and topological string theory. The derivation of the topological recursion formula suggests that it is probably the correct formulation of the Virasoro constraint condition for Calabi-Yau spaces.

References: We emphasize that this course serves as a mathematical introduction to the topics listed above. We intend to produce lecture notes to be published. As you can see, it would require an enormous amount of knowledge to understand everything listed below. Our plan is to provide a simple explanation of these fascinating subjects in this course. At least that's what I will try.

An excellent physics introduction to the material for mathematics audience is the Colloquium Talk on March 2, 2009, at UC Davis Mathematics Department by Vincent Bouchard. With his permission, I post his talk below.

Algebraic geometry of Hurwitz numbers, Hodge integrals, random matrix theory, representation of symmetric groups, and integrable systems.

Combinatorial theory of Hurwitz numbers. Topological recursion. Topological string theory and Hurwitz numbers. Recent developments.