Schedule

All talks will be held in 245 Altgeld Hall.

Saturday, March 14, 2009

8:30 am - Registration: Refreshments
9:30 am - Woodward
10:30 am - Break
11:00 am - Usher
12:00 noon - Lunch
2:00 pm - Lipshitz
3:00 pm - Break
3:30 pm - Ng

Sunday, March 15, 2009

8:30 am - Morning refreshments
9:30 am - Zinger
10:30 am - Break
11:00 am - Parker

Titles and Abstracts

Chris Woodward
Title: Functoriality of Gromov-Witten invariants under symplectic quotients Abstract: Mirror theorems in the sense of Givental relate generating functions for Gromov-Witten theory for the symplectic quotient of a vector space by a torus and the gauged Gromov-Witten theory for the action. I will describe how a geometric approach of Gaio-Salamon and Ziltener naturally leads to a notion of "morphisms of cohomological field theories" relating Gromov-Witten invariants of symplectic quotients to the equivariant Gromov-Witten theory, in general. Givental-type results are then special cases of (in general, conjectural) "quantum non-abelian localization".

Michael Usher
Title: Applications of filtration-theoretic invariants in Floer homology to symplectic topology
While the Floer homology associated to a Hamiltonian function on a symplectic manifold is independent of the Hamiltonian, the underlying Floer chain complex carries a real-valued filtration which is rather sensitive to the Hamiltonian. There have been some recent successes in using invariants extracted from this filtration to prove new results in symplectic topology and Hamiltonian dynamics. I'll discuss two such invariants: the "spectral invariant" of Schwarz and Oh, which leads to a sharp version of the inequality between a set's Hofer-Zehnder capacity and its displacement energy; and a new invariant called the "boundary depth," which leads to some results about coisotropic submanifolds.

Robert Lipshitz
Title: Bordered Floer homology: properties and computations
Heegaard Floer homology is an analogue of Seiberg-Witten theory defined using holomorphic curves. We will discuss an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with parametrized boundary. The talk will focus on the formal structure of the theory, how to compute it, and what those computations buy. This is joint work with Peter Ozsvath and Dylan Thurston.

Lenny Ng
Legendrian Symplectic Field Theory
Symplectic Field Theory is a package developed by Eliashberg-Givental-Hofer that counts holomorphic curves in symplectic manifolds. SFT has been surprisingly difficult to develop in the setting of Legendrian knots in contact manifolds, though a portion (contact homology) has been well-known for some time. I will report on recent progress toward formulating Legendrian SFT, involving a successful formulation for rational holomorphic curves in standard contact three-space, and speculate about possible applications to contact topology.

Aleksey Zinger
Title: Gromov-Witten invariants and integer curve counts
Gromov-Witten invariants of a compact symplectic manifold are certain virtual counts of J-holomorphic curves. These rational numbers are rarely integer, but are generally believed to be related to some integer counts. In string theory, these counts are known as instaton numbers and BPS states. The predictions of Aspinwall-Morrison and Gopakumar-Vafa for the existence of BPS states of Calabi-Yau 3-folds are extended by Pandharipande to all 3-folds, by Klemm-Pandharipande to all Calabi-Yau varieties in genus 0 and Calabi-Yau 4-folds in genus 1, and by Pandharipande and the speaker to Calabi-Yau 5-folds in genus 1. The last extension came as a bit of a surprise to some string theorists, who also feel that extensions to higher dimensions are impossible. The aim of this talk is to survey the known predictions, indicating how they arise, how the 6-dimensional case differs from the low-dimensional cases, and why they hold for Fano classes in 3-folds (symplectic manifolds of real dimension 6).

Thomas H. Parker
Title: Relating Gromov-Witten invariants of curves and surfaces
Little is known about the GW invariants of Kahler surfaces of general type. But remarkably, one can express the GW invariants of surfaces with p_g > 0 as sums of certain symplectic "local GW invariants" associated to spin curves. I'll describe my joint work with Junho Lee establishing this connection and expressing the local invariants in terms of a Taubes obstruction bundle over the space of maps into curves. Along the way, I'll explain how we overcome analytic difficulties that arise because the space of maps is not a manifold, and explain why the next step, the computation of the Euler class, is not approachable by standard algebraic geometry techniques.