Tag - Symplectic geometry

Amitai Zernik: Open Gromov-Witten theory of (ℂP1,ℝP1) in all genera and Gromov-Witten Hurwitz correspondence

4th December, 2017

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to (ℂP1,ℝP1). For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

We conjecture that the higher genus theory exists and give an explicit fixed-point formula for it. We show that this formula behaves as expected in the closed and disk-map sector, and satisfies a domain decomposition property. We consider open branched covers and prove they satisfy analogous properties. This implies a Gromov-Witten Hurwitz correspondence extending part of the work of Okounkov and Pandharipande from the closed to the open setting.

In the talk I'll discuss some of these ideas. I'll try to make the talk as self-contained as possible, and give examples.

Yu-Shen Lin: Open Gopakumar-Vafa conjecture for rational elliptic surfaces

27th November, 2017

We will explain a definition of open Gromov-Witten invariants on the rational elliptic surfaces and explain the connection of the invariants with tropical geometry. For certain rational elliptic surfaces coming from meromorphic Hitchin system, we will show that the open Gromov-Witten invariants with boundary conditions near infinity (up to some transformation) coincide with the closed geodesic counting invariants defined by Gaiotto-Moore-Neitzke, which are integer-valued.

Zhengyi Zhou: Morse-Bott cohomology from homological perturbation

6th November, 2017

Abstract: In this talk, I will give a new construction of the Morse-Bott cochain complex, where the underlying vector space is generated by the cohomology of the critical manifolds. This new construction has two nice features: (1) It requires the minimum amount of transversality. (2) The choices made in the construction do not depend on the moduli spaces. I will explain its relation to three other constructions in literature, namely Austin-Braam's push-pull construction, Fukaya's push-pull construction and the cascades construction. I will then discuss the equivariant counterpart and the generalization in the polyfold theory.

Yuan Gao: Wrapped Fukaya categories and functors

23rd October, 2017

Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated. In this talk, the story concerning the fundamental construction of Fourier-Mukai type functors of wrapped Fukaya categories is discussed, under slightly modified framework of wrapped Floer theory. Applications of the relevant techniques to be presented include the Kunneth formula and restriction maps.

Yoel Groman: Wrapped Floer theory and homological mirror symmetry for toric Calabi-Yau manifolds

9th October, 2017

Consider a Lagrangian torus fibration à la SYZ over a non-compact base. Using techniques from this arXiv paper, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity. The construction allows the formulation of a version of the homological mirror symmetry conjecture for open manifolds which are not exact near infinity. According to time constraints, I will apply this to prove homological mirror symmetry in the case where the A-model is the complement of an anti-canonical divisor in a toric Calabi Yau manifold.

Timothy Perutz: Floer theory in spaces of stable pairs over Riemann surfaces

4th May, 2017

I will report on joint work with Andrew Lee, which explores the notion that spaces of stable pairs over Riemann surfaces (in the sense of Bradlow and Thaddeus) could form a natural home for a 'non-abelian' analogue of Heegaard Floer homology for 3-manifolds - just as the g-fold symmetric product is the home of Heegaard Floer homology - thereby circumventing the problems with singularities that beset instanton-type theories. In an initial foray into this area, we set up a theory not for Heegaard splittings but for fibered 3-manifolds, based on fixed-point Floer homology. We show that, when the fiber has genus 1, it contains the expected information from the Seiberg-Witten Floer theory of the fibered 3-manifold.

Douglas Schultz: Lagrangian Floer theory in symplectic fibrations

27th April, 2017

Given a fibration of compact symplectic manifolds and an induced fibration of Lagrangians, one can ask if we can compute the Floer cohomology of the total Lagrangian from information about the base and fibre Lagrangians. The primary example that we have in mind is the manifold of full flags in ℂ3 which fibres as P1→Flag(ℂ3)→P2, and a Lagrangian T3 that fibres over the Clifford torus in P2. It turns out that one can prove the usual transversality and compactness results when the base is a rational symplectic manifold and the fibres are monotone. Assuming that we have a solution to the Maurer-Cartan equation, we then write down a Leray-Serre type spectral sequence which computes the Floer cohomology of the fibered Lagrangian. In the special case that the fibers are Kähler, we derive a formula for the leading-order disk potential.

Richard Siefring: Symplectic field theory and codimension-2 stable Hamiltonian submanifolds

20th April, 2017

Motivated by the goal of establishing a 'symplectic sum formula' in symplectic field theory, we will discuss the intersection behavior between punctured pseudoholomorphic curves and symplectic hypersurfaces in a symplectization. In particular we will show that the count of such intersections is always bounded from above by a finite, topologically determined quantity even though the curve, the target manifold, and the symplectic hypersurface in question are all non-compact.

Sheng-Fu Chiu: Sheaves and contact non-squeezing in ℝ2n × S1

13th April, 2017

In this talk I will introduce a way to associate a triangulated category of sheaves with a domain of ℝ2n×S1. The cohomological information on the category side helps to detect the contact non-squeezing property of the domain on the topology side.