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.
- Necessary
background: MAT 125AB, MAT 145, MAT 150ABC and MAT 185A, or something equivalent.
- Sufficient
(but unnecessary!) background:
MAT 201AB, MAT 205, MAT 215ABC, MAT 239, MAT 240AB
and MAT 250AB.
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.
- V. Bouchard, New formulae for Gromov-Witten invariants and Hurwitz numbers, UC Davis Mathematics Colloquium Talk, March 2, 2009.
Algebraic geometry of Hurwitz numbers, Hodge integrals, random matrix theory, representation of symmetric groups, and integrable systems.
- A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I, Proc. Symposia Pure Math. 80, 325--414 (2009).
- A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7, 447 (2000) [arXiv:math.AG/0004128].
- T. Graber and R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135, 25--36 (2003).
- C.-C.M. Liu, Formulae of one-partition and two-partition Hodge integrals, Geom. & Top. Monographs 8, 105--128 (2006).
- C.-C.M. Liu, K. Liu, and J. Zhou, A proof of a conjecture of Marino-Vafa on Hodge integrals, J. Diff. Geom. 65, 289--340 (2003).
- M. Kazarian and S. Lando, An algebro-geometric proof of Witten's conjecture, J. Amer. Math. Soc. 20, 1079--1089 (2007).
- J. Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, arXiv:0308024 math.AG (2003).
- I.P. Goulden and D.M. Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125, 51--60 (1997).
- I.P. Goulden, D.M. Jackson and R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc. 83:3, 563--581 (2001).
- B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1, 347--452 (2007).
- M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167, 179--222 (2007).
- M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20, 1--23 (2007).
- M. Mulase and B. Safnuk, Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy, Indian Journal of Mathematics 50, 189--228 (2008).
- K. Liu and H. Xu, Recursion formulae of higher Weil-Petersson volumes, Intern. Math. Res. Notices 2009, 835--859 (2009).
- V. Bouchard and M. Marino, Hurwitz numbers, matrix models and enumerative geometry, Proc. Symposia Pure Math. 78, 263--283 (2008).
- V. Bouchard, A. Klemm, M. Marino, and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287, 117--178 (2008).
- G. Borot, B. Eynard, M. Mulase, and B. Safnuk, A matrix model for simple Hurwitz numbers, and topological recursion, Preprint arXiv:0906.1206 physics.math-ph (2009).
- B. Eynard, M. Mulase, and B. Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers, Preprint arXiv:0907.5224 math.AG (2009).
- M. Mulase and N. Zhang, Polynomial recursion formula for linear Hodge integrals, Preprint arXiv:0908.2267 math.AG (2009).