# Seminars & Events for Symplectic Geometry Seminar

##### GIT and μ-GIT

**Please note different day and time. **In this lecture I will explain the moment-weight inequality, and its role in the proof of the Hilbert-Mumford numerical criterion for μ-stability. The setting is Hamiltonian group actions on closed Kaehler manifolds. The major ingredients are the moment map μ and the finite dimensional analogues of the Mabuchi functional and the Futaki invariant. This is joint work with Valentina Georgoulas and Joel Robbin, based on conversations with Xiuxiong Chen, Song Sun, and Sean Paul.

##### Positive loops and orderability in contact geometry

**PLEASE NOTE DIFFERENT TIME. ** Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional Σ so that the result is always orderable. The main technical tool is a Morse-Bott Floer theoretic growth rate, which has some parallels with Givental’s nonlinear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(Σ).

##### Finite Energy Foliations and Connect Sums

I will present some recent joint work with Richard Siefring on the behavior of finite energy foliations under the action of a 0-surgery (i.e. a connect sum) and a 2-surgery. We will then discuss applications to the restricted three body problem.

##### How not to define cylindrical contact homology

We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. One can fix this difficulty by using S1-dependent almost complex structures, but at the expense of introducing another difficulty which we will explain. We outline how fixing the latter difficulty ultimately leads to a different theory, an analogue of positive symplectic homology. This talk is intended to be part of a series of expository talks on the foundations of contact homology, but prerequisites should be minimal.

##### Enumeration of real rational curves

The classical problem of enumerating rational curves in projective spaces is solved using a recursion formula for Gromov-Witten invariants. In this talk, I will describe a similar relation for real Gromov-Witten invariants with conjugate pairs of constraints. An application of this relation provides a complete recursion for counts of real rational curves with such constraints in odd-dimensional projective spaces. I will outline the proof and discuss some vanishing and non-vanishing results. This is joint work with A. Zinger.

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

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.

##### Tori in four-dimensional Milnor fibres

The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. Time allowing, I will use these to give examples of fibres whose Fukaya categories are not generated by vanishing cycles.

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

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.

##### Gopakumar-Vafa conjecture for symplectic manifolds

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. In this talk, based on joint work with Tom Parker, I describe the proof of this conjecture, coming from a more general structure theorem for the Gromov-Witten invariants of symplectic 6-manifolds.

##### Feynman categories, universal operations and master equations

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. These give rise to Gerstenhaber's famous bracket, the pre-Lie structure of string topology, as well as to the Lie bracket underlying the three geometries of Kontsevich built from symplectic vector spaces.

##### Lagrangian submanifolds of complex projective space

First, I will discuss a proof that a Lagrangian torus in ℂℙ2 arising from a semitoric system described by Weiwei Wu coincides with the image in ℂℙ2 of Chekanov's exotic Lagrangian torus in ℝ4. I will then turn to what can be regarded as higher-dimensional versions of Wu's torus, which include a monotone Lagrangian torus in ℂℙ3 which is not isotopic either to the Clifford torus or to any of Chekanov and Schlenk's twist tori, as well as monotone Lagrangian submanifolds of ℂℙn for n at least 4 which (unusually for monotone Lagrangians) are Hamiltonianly displaceable. This is joint work with Joel Oakley.

##### A remark on the Euler equations of hydrodynamics

**Please note special day and time. **The time evolution of an ideal incompressible fluid is described by the Euler equations. In this mostly speculative talk I will discuss a connection between stationary solutions of these equations and symplectic topology, as well as possible applications to questions of hydrodynamic instability.

##### Cylindrical contact homology as a well-defined homology?

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. In addition we sketch a new approach allowing us to compute cylindrical contact homology for a large class of examples which admit contact forms that are admissible under the stronger conditions required. This approach is applicable to prequantization spaces and the links of simple singularities.

##### On Floer cohomology and non-archimedian geometry

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.

##### A criterion for generating Fukaya categories of fibrations

The Fukaya category of a fibration with singularities W: M --> 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. The new ingredients needed include a Floer homology group associated to (M,W) and the Serre functor. This is work in progress with Mohammed Abouzaid.

##### Implicit atlases and virtual fundamental cycles

An implicit atlas on a (moduli) space consists of certain auxiliary (moduli) spaces satisfying a precise set of axioms. We will summarize the construction of implicit atlases on moduli spaces of J-holomorphic curves, under the assumption of a precise "strong gluing" theorem. We will also describe an algebraic "theory of virtual fundamental cycles" (which does not use perturbation) in the abstract setting of spaces equipped with implicit atlases. This "VFC package" is sufficient to define Floer-type homology theories from a collection of (moduli) spaces equipped with a compatible system of implicit atlases.

##### Contact invariants in sutured monopole and instanton homology

**Please note special day (Wednesday) and time. **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. This is joint work with John Baldwin.

##### New combinatorial computations of embedded contact homollogy

Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and the almost complex structure on its symplectization. This fact can be used to extract symplectic geometric information (e.g. ECH capacities) but explicit computation of the chain complexes has been carried out only on a few cases. Extending the work of Hutchings-Sullivan, we combinatorially describe the ECH chain complexes of T^3 with general T^2 -invariant contact forms and certain almost complex structures.

##### Knot contact homology and topological strings

We describe the recently observed relation between knot contact homology and open topological strings. This in particular gives two ways of looking at the augmentation variety from knot contact homology, which describe the relevant string theory at the level of the disk. We discuss a generalization of knot contact homology in the spirit of full symplectic field theory that corresponds to full quantum string theory and conjecture a relation to the recursion relations for the colored HOMFLY polynomial.

##### TBA - Solomon

**Please note: two speakers on March 14.**