Schedule:

 May 16 - 20，2016

Mon, Tue, Wed morning 9 am -11 am : Jake Solomon

Thu, Fri morning 9 am -11 am : Sara Takachinsky

Mon, Tue, Wed, Fri afternoon 2 pm -4pm : Felix Janda

There is a 10-min break at the middle of each session (up to the speaker).

Titles & Abstracts:

1. Felix Janda

Title:

Tautological relations on the moduli space of curves

Abstract:

The tautological ring is a subring of the Chow (or cohomology) ring of the moduli space of curves containing many natural geometric classes. There is a canonical set of generators of the tautological ring, and the relations between the generators are called tautological relations.

In these series of lectures we first review the definition and the basic properties of the tautological ring of the moduli space of stable curves. We then proceed to review conjectures about the structure of the tautological ring and methods of producing tautological relations. The notion of a cohomological field theory with its applications to the recent advances in the study of tautological relations will play a central role in the lectures. In the end we will discuss recent work (joint with E. Clader) on tautological relations related to the double ramification cycle.

2. Jake Solomon

Title:

Stable disks, open KdV and Virasoro

Abstract:

I will discuss intersection theory on the moduli space of stable marked disks and how it leads to the open KdV hierarchy and the open Virasoro constraints. Canonical boundary conditions will be constructed that give rise to well-defined open descendent integrals. Hands-on examples will be presented. The open descendent integrals will be shown to satisfy the open string, dilaton and TRR equations. These equations allow the computation of all genus zero descendent integrals, motivating the definition of open KdV and Virasoro.

These talks are based on joint work with R. Pandharipande and R. Tessler.

3. Sara Tukachinsky

The goal of the talks is to define genus zero open Gromov-Witten invariants in arbitrary dimension. These invariants count stable disk maps with Lagrangian boundary conditions.

Following the approach of Fukaya et al., we will give A_\infty structures to the ring of differential forms on a Lagrangian submanifold. Then we define bounding chains and a natural equivalence relation on them. We give a canonical parametrization of the space of equivalence classes of chains. Next we show that the superpotential gives rise to a well-defined function on equivalence classes of bounding chains.  This function is the generating function of genus zero open Gromov-Witten invariants.

This is joint work with Jake Solomon.

Organizers: