**Title:** Real Gromov-Witten Theory In Genus Zero and Beyond

**Speaker:** Mohammad Farajzadeh Tehrani, Princeton

**Date:** Thursday, May 30, 2013

**Time:** 1:30pm – 2:30pm

**Place:** Lecture Hall 102, Simons Center

[box, type=”download”]Watch the video.[/box]

**Abstract** There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an antisymplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this talk, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We then extend this idea to genus one to define real g=1 invariants and talk about the difficulties of generalizing this construction to higher genus. Using equivariant localization over odd dimension projective space, I will provide several examples in genus zero and one, supporting and explaining this construction.

**Title:** Cotangent Bundles of Open 4-manifolds

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Adam Knapp, Columbia University

**Date:** Thursday, November 29, 2012

**Time:** 1:00pm – 2:00pm

**Place: **Seminar Room 313, Simons Center

**Abstract:** Using results of Eliashberg and Cieliebak, I show that: if X1 and X2 are two homeomorphic open 4-manifolds, then their cotangent bundles are symplectomorphic. As a corollary, all exotic R4s smoothly embed in the standard symplectic R8 as Lagrangian submanifolds.

Watch the Video.

Note: Due to technical difficulties the first 10 minutes of this talk are not available.

]]>Note: Due to technical difficulties the first 10 minutes of this talk are not available.

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**Title:** Refined Chern-Simons Theory and Hilbert Schemes of Points on the Plane

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Hiraku Nakajima, RIMS Kyoto University

**Date:** Thursday, October 25, 2012

**Time:** 11:30am – 12:30pm

**Place: **Seminar Room 313, Simons Center

**Abstract:** Aganagic-Shakirov recently propose a refined Chern-Simons theory which gives a one-parameter deformation of quantum invariants of 3-manifolds. A key ingredient is a deformation of the S-matrix acting on the quantum Hilbert space of 2-torus, which is a modification of an earlier proposal by Gukov, Iqbal, Kozcaz and Vafa. I will give an interpretation of the deformed S-matrix in terms of Hilbert schemes of points on the plane.

[box, type=”download”]Watch the video.[/box] ]]>

**Title:** A Bordered Monopole Floer Theory

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** John Baldwin, Boston College

**Date:** Thursday, October 11, 2012

**Time:** 1:00pm – 2:00pm

**Place:** Seminar Room 313, Simons Center

**Abstract:** I’ll discuss work-in-progress toward constructing monopole Floer theoretic invariants of bordered 3-manifolds. Roughly, our construction associates an A-infinity algebra to a surface, an A-infinity module to a bordered 3-manifold, and a map of A-infinity modules to a 4-dimensional cobordism of bordered 3-manifolds. I’ll focus on the topological and algebraic aspects of our work and, in particular, will indicate how we prove a pairing theorem relating the invariants of two bordered 3-manifolds with that of the manifold obtained by gluing the former together along homeomorphic components of their boundaries. This is joint work with Jon Bloom.

**Title:** Szabo’s Geometric Spectral Sequence for Tangles

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** John Baldwin, Boston College

**Date:** Thursday, October 11, 2012

**Time:** 10:15am – 11:15am

**Place:** Seminar Room 313, Simons Center

**Abstract:** Szabo recently wrote down (seemingly out of nowhere) a combinatorial chain complex associated to a link diagram L in S^3, modeled on Khovanov homology, which conjecturally computes the Heegaard Floer homology of the branched double cover of S^3 along L. In this talk, I’ll describe an extension of Szabo’s work to tangles, modeled on Khovanov’s “functor-valued invariant of tangles.” I’ll introduce A-infinity algebras and modules and describe how to compute Szabo’s invariant of a link L by decomposing L into tangles and computing the invariants of these simpler objects (our pairing theorem). At the end, I’ll mention a striking relationship (coincidental?) between the structures apparent in this tangle theory and a version of the Fukaya category of the once-punctured torus explored by Lekili and Perutz. This talk will be fairly basic and should provide intuition for the more complicated construction I’ll describe in the Topology Seminar. This is joint work with Cotton Seed.

**Title:** Uniqueness of the Approximating Contact Structure

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Thomas Vogel, Max Planck Institue for Mathematics in Bonn, Germany

**Date:** Tuesday, October 9, 2012

**Time:** 11:15am – 12:15pm

**Place:** Seminar Room 313, Simons Center

**Abstract:** A theorem of Eliashberg and Thurston implies that every foliation on a 3-manifold can be approximated by a contact structure. In this talk, we show that in many cases the contact structure obtained in this way is unique up to isotopy and give apply this fact to show that certain spaces of taut foliations are not connected.

**Title:** Odd Khovanov Homology Via Hyperplane Arrangements

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Zsuzsanna Dancso, Fields Institute

**Date:** Thurssday, October 4, 2012

**Time:** 1:00pm – 2:00pm

**Place:** Seminar Room 313, Simons Center

**Abstract:** We will describe a construction of Odd Khovanov homology (isomorphic to that of Ozsvath-Rasmussen-Szabo) as a special case of a homology theory for (signed) hyperplane arrangements. Hyperplane arrangements are a combinatorial structure which include graphs and link projections as subsets. Our construction is invariant under Gale duality, which is a notion of duality generalising planar graph duality. The talk is based on joint work with Anthony Licata; arXiv: 1205.2784.

[box, type=”download”]Watch the Introduction video.[/box]

[box, type=”download”]Watch the video.[/box]

]]>**Title:** From ECH to HF: The Story You Have Never Heard

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Paolo Ghiggini, Univ. de Nantes

**Date:** Tuesday, October 2, 2012

**Time:** 10:00am – 11:00am

**Place:** Seminar Room 313, Simons Center

**Abstract:** After describing definitions of \widehat{HF} and \widehat{ECH} adapted to open book decompositions, I will define a map from \widehat{ECH}(M) to \widehat{HF}(-M). This map is defined by counting J-holomorphic curves in a symplectic cobordism but, unlike the definition of the map from \widehat{HF} and \widehat{ECH}, the construction is subtle and seems related to a Floer theoretic implementation of Poincaré duality in (S1)^{2g}.

**Title:** Nonorientable Four-ball Genus Can be Arbitrarily Large

**Program:** Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

**Speaker:** Joshua Batson, MIT

**Date:** Thursday, September 27, 2012

**Time:** 1:00pm – 2:00pm

**Place: **Seminar Room 313, Simons Center

[box, type=”download”]Watch the Introduction video.[/box]

the orientable four-ball genus, which can be bounded using algebraic topology, Heegaard-Floer homology, and Khovanov homology, the best lower bound in the literature

on the nonorientable four-ball genus for any K is 3. We find a lower bound in terms of the signature of K and the Heegaard-Floer d-invariant of the integer homology

sphere given by -1 surgery on K. In particular, we prove that the nonorientable four-ball genus of the torus knot T(2k,2k-1) is k-1.

[box, type=”download”]Watch the video.[/box]

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