MAT 298: Seminars on Gromov-Witten Theory

Spring Quarter 2010. CRN = #69853

Fridays 4:00 - 6:00

3106 MSB

Organizer: Motohico Mulase

This term we meet on Fridays 4:10 to 6:00. However, several talks
are scheduled on other days or time slots due to the availability of speakers.
Please note the irregular meeting time below.

** April 2:
Motohico Mulase.** "Transcendence of π and *e*,
volume of polytopes, and Gromov-Witten invariants."

Analyzing the theorem of Hermite and Lindemann that

*e* and π are transcendental numbers, we notice that
they are transcendental because (1)

*e* is a special integer value of the exponential function
exp(

*x*) and 2π

*i* is a

*period* of exp(

*x*); (2) the exponential function
is defined as the unique solution of a differential equation ƒ′ = ƒ with the initial condition ƒ(0) = 1;
and (3) both equations
are defined over integers. Would the same kind of
phenomena happen if we consider a more general differential
equation, such as (℘′)

^{2}
= 4 ℘

^{3} −

*g*_{2}℘
−

*g*_{3} , when the coefficients

*g*_{2} and

*g*_{3} are
algebraic numbers? As an introduction to this year's
seminar, I will start with giving the elementary proof of
the transcendence of

*e* and π, and then relate it with the
idea of

*period* due to Kontsevich and
Zagier. The

*polynomiality* appearing in the
work of Mirzakhani, Norbery, Zhang and myself, and others
is what we
would like to understand.

** April 9:**
No Meeting. Please attend the two talks
by Dr. Xu scheduled in the following
week.

** Monday, April 12, 4:10 - 5:00 in MSB 1147 (Department Colloquium):
Hao Xu,
Harvard University.**
"Tautological Ring of Moduli Spaces of Curves."

We will survey the recent progress on the structure of tautological ring of moduli space of curves. We will talk about our proof (with Prof. Kefeng Liu) of the Faber intersection number conjecture and recursion formulae of higher Weil-Petersson volumes of moduli spaces of curves. The latter is motivated by the work of Mulase and Safnuk.

PDF file of Dr. Xu's Talk
(An excellent survey paper by Dr. Xu on these topics is

"Descendent integrals and tautological rings of moduli spaces of curves.")

** Tuesday, April 13, 4:10 - 5:00 in MSB 2112 (Geometry/Topology Seminar):
Hao Xu,
Harvard University.**
"Mirror symmetry of singularities."

The FJRW theory for nondegenerate quasihomogeneous singularities provides a mathematical framework for the Landau-Ginzburg/Calabi-Yau correspondence. We will discuss Witten's ADE hierarchy conjecture proved by Fan-Jarvis-Ruan and their relations with Gromov-Witten theory.

** April 16, 2:10 - 3:00 in 1147 MSB (RFG Seminar on Hurwitz numbers): Motohico Mulase. ** "From a combinatorial recursion to the topological recursion, Lecture 1."

Hurwitz numbers satisfy a set of combinatorial relations due to Hurwitz, i.e., the cut-and-join equations. Although these equations determine the Hurwitz numbers, the actual computation does not go very fast because one has to solve the simultaneous equations. In 2007 physicists Bouchard and Marino conjectured an effective recursion formula for Hurwitz numbers using a quite different generating function. Their conjectural formula is a special case of the "topological recursion" discovered by Eynard and Orantin in Random Matrix Theory. Computer experiments showed that the Bouchard-Marino formula gave correct Hurwitz numbers, but no mathematical proof was presented at that time. In a series of collaborations with Borot, Eynard, Safnuk, and Zhang, I have discovered that the Bouchard-Marino formula is the Laplace transform of the cut-and-join equation, and thus have proved the conjecture.

In this lecture, we outline the statement that the Bouchard-Marino formula is the Laplace transform of the cut-and-join equation.

** April 16, 4:10 - 6:00 in 3106 MSB: Frank Liou. ** "The Grothendieck Riemann-Roch theorem and the relations among tautological clases."

** April 23, 2:10 - 3:00 in 1147 MSB (RFG Seminar on Hurwitz numbers): Motohico Mulase. ** "From a combinatorial recursion to the topological recursion, Lecture 2."

In Lecture 2, we present a current research frontier where researchers are seeking the meaning of the Laplace transform. The Laplace transform in this context is indeed the "mirror symmetry" for toric Calabi-Yau threefolds. It changes the combinatorial structures among algebro-geometric objects into the symplectic invariants called the Gromov-Witten invariants. After this general conjectural vision is presented, we examine a concrete example. In this example, the main objective is counting the lattice points in orbifold polytopes. Amazingly, the Laplace transform of the numbers of these lattice points gives rise to the Witten-Kontsevich formula for intersection numbers of moduli spaces of stable curves! This fact is pointing to the combinatorial origin of the Virasoro constraint conditions.

Although we will not have the time to explain in these two lectures, the relation between the two different subjects of Lecture 1 (Hurwitz numbers) and Lecture 2 (lattice point counting) is a one parameter family of deformations of solutions of a completely integrable system of nonlinear partial differential equations called the KP equations. This explains the appearance of the KdV equation in the Witten-Kontsevich theory, and illustrates the Frobenius manifold structure of Dubrovin and the Givental formalism of integrable systems.

This deformation theory should be related to geometric deformation of Hurwitz covers and Belyi morphisms. Indeed, the lattice points are the set of Belyi morphisms. Most likely there is a combinatorial correspondence between Hurwitz covers and Belyi morphisms when the simple ramification points of the Hurwitz covers are set to be roots of unity.

** April 23, 4:10 - 6:00 in 3106 MSB: Naizhen Zhang. **

** Monday, April 26, in MSB 1147 (Department Colloquium):
Ravi Vakil,
Stanford University. **
"Generalizing the Cross Ratio: The Moduli Space of n Points on the Projective Line up to Projective Equivalence."

Four ordered points on the projective line, up to projective equivalence, are classified by the cross ratio, a notion introduced by Cayley. This theory can be extended to more points, leading to one of the first important examples of an invariant theory problem, studied by Kempe, Hilbert, and others. Instead of the cross ratio (a point on the projective line), we get a point in a larger projective space, and the equations necessarily satisfied by such points exhibit classical combinatorial and geometric structure. For example, the case of six points is intimately connected to the outer automorphism of

*S*_{6}. We extend this picture to an arbitrary number of points, completely describing the equations of the moduli space. This is joint work with Ben Howard, John Millson, and Andrew Snowden. This talk is intended for a general mathematical audience, and much of the talk will be spent discussing the problem, and an elementary graphical means of understanding it.

** April 30, 4:10 - 5:00 in 3106 MSB: Bertrand Eynard,
Institut de Physique Théorique, CEA-Saclay, France.**
"Introduction to the topological recursion and its application."

It was recently discovered (2004) that the coefficients

*F*_{g} in the large

*N* expansion of matrix integrals ln

*Z* = Σ

_{g} *N*^{ 2−2g} *F*_{g} , obey a certain recursion relation.
This recursion, and the coefficients

*F*_{g} given by it, can also be defined by themselves, independently of any possibly underlying matrix model, they depend solely on a plane curve, called the spectral curve.
The coefficients

*F*_{g} have many properties, they are invariant under symplectic transformations of the spectral curve, their deformations satisfy some special geometry, their limits commute with resolving singularities, they are almost modular forms (and can be easily turned into modular forms), and their sum is the τ-function of an integrable system.
They also have many applications, not only in random matrices. They count discrete surfaces (maps) of given genus, they count partitions and plane partitions, they count volumes of moduli spaces of Riemann surfaces (Mirzakhani's relations appear as a special case where the spectral curve is chosen to be

*y* = sin√

*x* ), and they are conjectured to give the Gromov-Witten invariants of every toric Calabi-Yau 3-fold when the spectral curve is the mirror curve of the Calabi-Yau manifold.

(An excellent survey paper by Professor Eynard on these topics is

"Geometrical interpretation of the topological recursion, and integrable string theories.")

** June 4, 4:10 - 6:00 in 3106 MSB: Brad Safnuk, Central Michigan University.**