# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### Algebraic structures associated to Weinstein manifolds

##### Symplectic Geometry of Quantum Noise

We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables in operational quantum mechanics.

##### Homological Mirror Symmetry for a Calabi-Yau hypersurface in projective space

We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. After stating the result carefully, we will describe some of the techniques used in its proof, and draw lots of pictures in the one-dimensional case.

##### On the symplectic invariance of log Kodaira dimension

Every smooth affine variety has a natural symplectic structure coming from some embedding in complex Euclidean space. This symplectic form is a biholomorphic invariant. An important algebraic invariant of smooth affine varieties is log Kodaira dimension. One can ask, to what extent is this a symplectic invariant? We show some partial symplectic invariance results for smooth affine varieties of dimension less than or equal to 3.

##### Hamiltonian S^1 Actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds

The question of what conditions guarantee that a symplectic circle action is Hamiltonian has been studied for many years. In 1998, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. In 2010, Cho, Hwang, and Suh proved in the 6-dimensional case that if we have b_2^+=1 at a reduced space at a regular level \lambda of the circle valued moment map, then the action is Hamiltonian. (Please click on seminar title for complete abstract.)

##### Behavior of Welschinger invariants under Morse simplification

Welschinger invariants, real analogs of genus 0 Gromov-Witten invariants, provide non-trivial lower bounds in real algebraic geometry. In this talk I will explain how to get some wall-crossing formulas relating Welschinger invariants of the same (up to deformation) rational algebraic surface with different real structures. This relation is obtained via a real version of a formula by Abramovich and Bertram which computes Gromov-Witten invariants using deformations of complex structures. It can also be seen as a real version, in our special case, of Ionel and Parker's symplectic sum formula. If time permits, I will give some qualitative consequences of this study, for example the vanishing of Welschinger invariants in some cases, and will discuss some generalizations. This is joint work with Nicolas Puignau (UFRJ, Rio de Janeiro)

##### An arithmetic refinement of homological mirror symmetry for the 2-torus

We establish a derived equivalence of the Fukaya category of the 2- torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z. It specializes to an equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the nodal Weierstrass curve y^2+xy=x^3, and, over the punctured disc Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2- torus. This is joint work with Tim Perutz.

##### Abstract analogues of flux as symplectic invariants

This talk is part of a circle of ideas that one could call "categorical dynamics". We look at how objects of the Fukaya category move under deformations prescribed by fixing an odd degree quantum cohomology class. This is an analogue of moving Lagrangian submanifolds under non-Hamiltonian deformations. It leads to a new invariant of closed symplectic manifolds, which can distinguish deformation equivalent symplectic structures.

##### A reverse isoperimetric inequality for J-holomorphic curves.

I'll discuss a bound on the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. (CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.)

##### Gromov-Witten theory and cycle-valued modular forms

A remarkable phenomenon in Gromov-Witten theory is the appearance of (quasi)-modular forms. For example, Gromov-Witten generating functions for elliptic curve, local $\mathbb{P}^2$, elliptic orbifold $\mathbb{P}^1$ are all (quasi)-modular forms. In this talk, we will discuss modularity property of the Gwomov-Witten cycles of elliptic orbifold $\mathbb{P}^1$. (CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.)

##### The Calabi homomorphism, Lagrangian paths and special Lagrangians

The talk will have two parts. First, I'll discuss a generalization of the Calabi homomorphism to a functional on Lagrangian paths. Then I'll explain how the functional relates to special Lagrangian geometry, and as time permits, how it fits into the framework of mirror symmetry.

##### The open Gromov-Witten-Welschinger theory of blowups of the projective plane

I'll explain how to compute the Welschinger invariants of blowups of the projective plane at an arbitrary conjugation invariant configuration of points. Specifically, open analogues of the WDVV equation and Kontsevich-Manin axioms lead to a recursive algorithm that reconstructs all the invariants from a small set of known invariants. This result is joint work with Asaf Horev.

##### Open-closed Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds

We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses

generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.

##### Arnold Conjecture for Clifford Symplectic Pencils

A symplectic pencil is a linear family of symplectic forms, i.e., a linear space of two-forms, each of which, except of course the zero form, is symplectic. Symplectic pencils arise, for instance, from representations of Clifford algebras and can be thought of as an analogue of the symplectic structure in one interpretation of the least action principle with multi-dimensional time. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Toric b-symplectic and origami manifolds

Origami manifolds and b-symplectic manifolds are examples of manifolds which are symplectic except on a hypersurface Z: in the origami case, the symplectic form vanishes at Z, in the b-case, it explodes to infinity at Z. In this talk we will translate Delzant's classification of toric symplectic manifolds by their moment images to these two extreme cases, and see what that tells us about toric origami manifolds and toric b-symplectic manifolds.**Ana Rita Pires**,*Cornell University*

##### Contact Non-Squeezing and Rabinowitz Floer Homology

We will present joint work with Will Merry. Using spectral invariants in Rabinowitz Floer homology we present an abstract contact non-squeezing theorem for periodic contact manifolds. We then exemplify this in concrete examples. Finally we explain connections to the existence of a biinvariant metric on contactomorphism groups. All this is connected and generalizes work by Eliashberg-Polterovich and Sandon.

##### Symplectic cohomology and loop homology

The string topology of Chas-Sullivan produces operations on the homology of the free loop space of orientable manifolds, and analogous structures are known to exist on the symplectic cohomology of their cotangent bundles. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Intermediate symplectic capacities

In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1<d<n. I will discuss the answer to this question and its relevance in symplectic geometry. This is joint work with San Vu Ngoc.

##### Lagrangian caps in high dimensional symplectic manifolds

I will present a recent result (joint with Yakov Eliashberg) demonstrating the existence of exact Lagrangian cobordisms with a loose Legendrian in the negative end, in all dimensions greater than 4. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Resonance for loop homology on spheres

Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space LM of M are the closed geodesics on M. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.