Joint IAS-PU Symplectic Geometry Seminar

Contact invariants in sutured monopole and instanton homology

Steven Sivek
University of Warwick
March 5, 2014
Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. In this talk, I will construct an invariant of a contact structure on a 3-manifold with boundary as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps and an exact triangle associated to bypass attachment, and explain how this construction leads to an invariant in the sutured version of instanton Floer homology as well.

A criterion for generating Fukaya categories of fibrations

Sheel Ganatra
Stanford University
February 21, 2014
The Fukaya category of a fibration with singularities \(W: M \to C\), or Fukaya-Seidel category, enlarges the Fukaya category of \(M\) by including certain non-compact Lagrangians and asymmetric perturbations at infinity involving \(W\); objects include Lefschetz thimbles if \(W\) is a Lefschetz fibration. I will recall this category and then explain a criterion, in the spirit of work of Abouzaid and Abouzaid-Fukaya-Oh-Ohta-Ono, for when a finite collection of Lagrangians split-generates such a fibration.

On Floer cohomology and non-archimedian geometry

Mohammed Abouzaid
Columbia University
February 14, 2014
Ideas of Kontsevich-Soibelman and Fukaya indicate that there is a natural rigid analytic space (the mirror) associated to a symplectic manifold equipped with a Lagrangian torus fibration. I will explain a construction which associates to a Lagrangian submanifold a sheaf on this space, and explain how this should be the mirror functor.

Cylindrical contact homology as a well-defined homology?

Joanna Nelson
Institute for Advanced Study; Member, School of Mathematics
February 7, 2014
In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop a theory without the use of virtual chains or polyfolds in 3 dimensions. It turns out that this can be accomplished by placing strong conditions on the growth rates of the indices of Reeb orbits.

Feynman categories, universal operations and master equations

Ralph Kaufmann
Purdue University; Member, School of Mathematics
December 6, 2013
Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g. open/closed versions or those appearing in homological mirror symmetry. For any such Feynman category there is an associated Feynman category of universal operations.

Calabi-Yau mirror symmetry: from categories to curve-counts

Tim Perutz
University of Texas at Austin
November 15, 2013
I will report on joint work with Nick Sheridan concerning structural aspects of mirror symmetry for Calabi-Yau manifolds. We show (i) that Kontsevich's homological mirror symmetry (HMS) conjecture is a consequence of a fragment of the same conjecture which we expect to be much more amenable to proof; and, in ongoing work, (ii) that from HMS one can deduce (some of) the expected equalities between genus-zero Gromov-Witten invariants of a CY manifold and the Yukawa couplings of its mirror.

Volume in Seiberg-Witten theory and the existence of two Reeb orbits

Daniel Cristofaro-Gardiner
University of California, Berkeley; Member, School of Mathematics
November 1, 2013

I will discuss recent joint work with Vinicius Gripp and Michael Hutchings relating the volume of any contact three-manifold to the length of certain finite sets of Reeb orbits. I will also explain why this result implies that any closed contact three-manifold has at least two embedded Reeb orbits.