WORKSHOP 2010

Abstracts

Tom Coates

« Gromov--Witten Invariants and Modular Forms »

Abstract: I will show that generating functions for higher-genus

Gromov--Witten invariants of certain non-compact Calabi--Yau 3-folds are

modular forms. Along the way, I will state a conjecture that gives a

simple and straightforward description of how Gromov--Witten invariants

(in all genera) change under crepant birational transformations; this

conjecture has been proved in a number of toric examples. This is joint

work with Hiroshi Iritani.

Dominic Joyce

« D-orbifolds, Kuranishi spaces and polyfolds »

Abstract

Several important areas in symplectic geometry involve moduli spaces of J-holomorphic curves, such as open and closed Gromov-Witten theory, Lagrangian Floer homology, and Symplectic Field Theory. To "count" moduli spaces of J-holomorphic curves -- form a virtual cycle or chain -- one must first put a suitable geometric structure on these moduli spaces. In the most general case, in which the moduli spaces can be singular, there are two competing descriptions of this geometric structure: the "Kuranishi spaces" of Fukaya-Oh-Ono-Ohta, and the "polyfolds" of Hofer-Wysocki-Zehnder. The philosophies are opposed: in principle Kuranishi spaces remember only the minimal amount of geometric information on the moduli space needed to form a virtual cycle, whereas polyfolds remember essentially everything about the differential geometric moduli problem which defined the moduli space. So one expects a truncation functor going from polyfolds to Kuranishi spaces.

Unfortunately there are problems with a lot of analytic foundations of the FO3 picture, including the definition of Kuranishi spaces: none of the definitions of Kuranishi space given so far have been satisfactory. I will explain work in progress on what I believe to be the "correct" definition of Kuranishi space, which draws on the Derived Algebraic Geometry programme of Jacob Lurie and others, and a recent paper arXiv:0810.5174 on "derived manifolds" by Lurie's student David Spivak. Roughly speaking, we want a Kuranishi space to be a "derived orbifold with corners". However, I use a truncated, simplified version of Spivak's construction, so that my kind of derived orbifolds ("d-orbifolds") are a 2-category, whereas Spivak's derived manifolds are an infinity category (a simplicial category).

I expect there will be a truncation functor from HWZ polyfolds to d-orbifolds; also, there should be a nicely behaved virtual chain/virtual cycle construction for oriented d-orbifolds, using either the old FO3 technology or my ideas on "Kuranishi (co)homology". The Kuranishi cohomology approach does not involve perturbation of moduli spaces, and will simplify FO3 style Lagrangian Floer cohomology a great deal.

Pierre Schapira

« From the microlocal theory of sheaves to the Fukaya category »

Abstract:

The microlocal theory of sheaves has been introduced by Masaki Kashiwara and the

speaker in the 80's. This theory is associated to the homogeneous symplectic structure of the cotangent bundle to real manifolds and treat conic co-isotropic

submanifolds. Recently, Dmitry Tamarkin, by adapting the tools of this theory to

the non conic case, has obtained results of non-displaceability which suggest

that the category he constructed is closely related to the Fukaya category.

In this talk we will review these topics and explain some conjectures elaborated

jointly with Stéphane Guillermou.

Jacob Rasmussen

« Holomorphic triangles and maps induced by contact structures »

Abstract:

One natural way to define a map between two Lagrangian Floer homology

groups is to embed the Floer chain complex of the domain into the

chain complex of the target group. Another is by counting

pseudoholomorphic triangles. I'll discuss how these two approaches are

related in the context of Honda-Kazez-Matic's maps on sutured Floer

homology.

Sheila Sandon

« Application of generating functions to contact rigidity phenomena in R^2n x S1 »

Abstract

In his 1992 article on generating functions Viterbo defined invariants c^+ and c^- for compactly supported Hamiltonian symplectomorphisms of R^2n. The definition of c^+ and c^- can easily be generalized to the case of the contact manifolds R^2n+1 and R^2n x S1. However, while in the symplectic case c^+ and c^- are invariant by conjugation, in the case of R^2n+1 and R^2n x S1 only respectively the vanishing and the integer part of them are invariant. In my talk I will try to explain the reason of this crucial difference. I will also discuss several applications of c^+ and c^- in the case of R^2n x S1: orderability of R^2n x S1 (following Bhupal), the construction of an integer valued bi-invariant metric on the contactomorphism group and of an integer valued contact capacity for domains, and a new proof of the contact non-squeezing theorem of Eliashberg, Kim and Polterovich. If there will be time I will also discuss how to generalize to R^2n x S1 Traynor's construction of symplectic homology and how to use an equivariant version of it to reprove orderability of Lens spaces (Milin 2008).