Geometry and Physics Seminar
4-6pm in 4088 East Hall

Sep 10: Eric Zaslow (Northwestern)
Constructible Sheaves and the Fukaya Category
Abstract
I will describe an equivalence between these two categories, focusing on simple examples. The talk is based on work of Nadler and joint work with Nadler.
Sep 17: Aleksey Zinger (StonyBrook)
The Geometry of Genus-One Gromov-Witten Invariants and Mirror Symmetry
Abstract
The mirror symmetry principle of string theory has led to astounding predictions for counts of holomorpic curves. The verification of the original 1991 prediction for genus-0 GW-invariants of a quintic threefold (Q3) in the mid 1990s was quickly followed by proofs of MS formulas for genus-0 invariants of other manifolds. On ther other hand, the 1993 genus-1 BCOV prediction for Q3 (and other positive-genus MS formulas) remained elusive until recently. I will describe geometric properties of genus-1 invariants that make them about as computable as genus-0 invariants. In particular, they lead to the proof of the genus-1 BCOV prediction for Q3, by applying the classical localization theorem. The first hour will consist of a general overview of everything involved, except for the localization part. In the second hour, I am planning to discuss the localization computation, which in a sense is a reduction to genus 0, but I'd be happy to talk about details of any of the geometric properties if this is preferred by the audience.
Sep 24: Albrecht Klemm (Wisconsin)
Topological String Theory on Compact Calabi-Yau
Abstract
We describe the recent progress which lead to a solution of topological string theory on the quintic up to genus 51. We address all genus results on K3 fibred Calabi-Yau spaces in the fibre direction. We discuss ideas from matrix theory which could lead to a complete solution of topological string theory on compact Calabi-Yau.
Oct 1: Paul Johnson (Michigan)
The Equivariant Gromov-Witten Theory of stacky P^1s and Integrable Hierarchies
Abstracat
The Gromov-Witten invariants of a space give a large number of invariants, which have many relations between them. In some cases, these relations are especially nice: the generating function of the Gromov-Witten invariants is a solution of an integrable hierarchy - a family of infinitely many commuting differential operators. We describe recent work showing that the Equivariant Gromov-Witten invariants of stacky P^1s satisfy commuting copies of the 2-Toda hierarchy.
Oct 8: Renzo Cavalieri (Michigan)
Gerby localization and the GW theory of C^3/Z_3
Abstract
In this talk we will discuss a successful approach to the computation of orbifold Gromov-Witten invariants. Such invariants are interpreted in terms of G-Hodge integrals. Relations among G-Hodge integrals are obtained via Atiyah-Bott localization. However, in order to find sufficiently many relations, we will turn our attention to moduli spaces of maps to gerbes over P^1.
Oct 15: Fall Break
Oct 22: Vincent Bouchard (Harvard)
Remodeling the B-model: new ideas in enumerative geometry
Abstract
Recently we proposed a new, complete formalism to compute B-model open and closed topological string amplitudes in local Calabi-Yau geometries, including the mirrors of toric Calabi-Yau threefolds. The formalism is non-perturbative in the moduli, hence can be used to study various phases in the open/closed moduli space, such as orbifold points. In this talk I would like to summarize our B-model formalism, and focus on some of its mathematical implications, leading to new ideas/conjectures in orbifold enumerative geometry and Hurwitz theory.
Oct 29: Igor Dolgachev (Michigan)
McKay's correspondence for cocompact discrete subgroups of $SU(1,1)$
Nov 5: Sheldon Katz (UIUC)
Deformed quantum cohomology
Abstract
A generalization of quantum cohomology is suggested by half-twisting (0,2) models in string theory. The input data is a compact Kahler manifold M and a vector bundle E satisfying c_i(E) = c_i(M) for i=1,2. If E is the tangent bundle of M, ordinary quantum cohomology results. The talk will review quantum cohomology, explain what is known and what is not known about this suggested generalization of quantum cohomology, and give a glimpse into the physical origins. The main examples will be when E is a deformation of the tangent bundle, resulting in a Frobenius algebra depending on the deformation parameters which deform the usual quantum cohomology ring. This talk is based on joint works with Eric Sharpe and Josh Guffin.
Nov 12: Mark Gross (UCSD)
Tropical geometry and mirror symmetry
Abstract
I will explain some recent developments in understanding both sides of the mirror symmetry story in terms of tropical geometry. Starting from an integral affine manifold with singularities (i.e. a real manifold with coordinate charts off of a codimension two set, all of whose transition maps are integral affine linear), one can build a mirror pair of Calabi-Yau manifolds. Tropical data on the affine manifold (data of a piecewise linear nature) is related to holomorphic curves on one of the Calabi-Yau manifolds and period information on the other one. This picture is still largely conjectural, but already leads to some new and testable enumerative predictions counting certain kinds of curves on weight projective planes.
Nov 19: Pan Peng (Harvard)
On a proof of the Labastida-Marino-Ooguri-Vafa conjecture
Abstract
There have been a lot of marvelous results revealed by string theory, which deeply relate different aspects of mathematics. All these mysterious relations are connected by a core idea in string theory called "duality". Based on large $N$ Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants in the topological string theory. In this talk, I will discuss a proof of this conjecture and its relation to other problems, for example, the famous volume conjecture.
Dec 3: David Nadler (Northwestern)
Springer theory via the Hitchin fibration
Abstract
The geometry of the adjoint quotient g/G plays a central role in representation theory. I will describe how it is easy to see certain aspects of its structure in the symplectic geometry of the cotangent bundle T^*(g/G). In particular, the Springer theory of Weyl group representations arises as a toy model of constructions arising in the geometric Langlands program.
Dec 10: David Ben-Zvi (Texas)
Langlands Duality and Topological Field Theory
Abstract
I will describe a new framework for the representation theory of real and complex semisimple Lie groups, developed in joint work with David Nadler. We show how different aspects of representation theory (including the theory of intertwining operators, Lusztig's character sheaves and character varieties for surface groups) fit into the structure of a topological field theory (the character theory) which we associate to a Lie group. The main result is an equivalence between the character theories for Langlands dual groups, which may be considered as a three-dimensional aspect of the electric-magnetic duality of four-dimensional gauge theories. Time permitting, I will discuss how our approach gives a strong form of the local Langlands program over the reals.
Dec 17: Alessandro Chiodo
Twisted Gromov--Witten $r$-spin potential and Givental's quantization
Abstract
Mumford used the Grothendieck Riemann--Roch formula to express the Chern characters of the Hodge bundle over the moduli space of stable curves via other tautological classes. Faber and Pandharipande generalized his result on spaces of stable maps. They showed that the resulting formula allows one to express the twisted Gromov--Witten potential (enriched with Chern characters of the Hodge bundle) of any target K{\"a}hler manifold via the usual Gromov--Witten potential. Givental found that the above result admitted a strikingly concise formulation in the framework of his quantization formalism. In this talk I will present work done in collaboration with Dimitri Zvonkine, where we generalize all these steps for the spaces of r-spin structures and maps.
Jan 28: Joe Harris (Harvard)
TBA