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Symplectic and Contact Geometry and Connections to Low-Dimensional Topology

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Organized by Peter Ozsváth and Yakov Eliashberg
Fall 2012 and Spring 2013

In the last 2 decades, and especially in recent years there were symplectic geometric ideas and methods brought significant progress in low-dimensional topology, while the methods developed in 3- and 4-dimensional topology found applications to symplectic geometric problems.

For instance, the Lagrangian intersection theory, and in particular, Lagrangian Floer homology theory which was created for symplectic-geometric needs led to the creation of Heeggard homology theory. In turn, Heegaard homology theory led to discovery of new invariants of transversal and Legendrian knots in contact 3-manifolds. In a different direction, amazingly effective invariants of knots can be obtained by studying Legendrian knot invariants of their Legendrian lifts to the unit cotangent bundle, which is a 5-dimensional contact manifold. Furthermore, the ideas explored in symplectic field theory were important in the creation and development of bordered Heegaard homology theory, the extension of Heegaard Floer homology to three-manifolds with boundary. Another construction using
symplectic methods, the embedded contact homology theory, played a crucial role in the recent proof of equivalence between Heegaard homology, embedded contact homology, and Seiberg-Witten Floer homology for 3-manifolds.  A promising current development in the symplectic topology of high-dimensional Weinstein manifolds and their contact boundaries stems from ideas borrowed from embedded and Heegaard homology theories in dimension 3.

Speaker and Seminar Schedule:

The weekly talks take place Thursdays at 1:00 in room 313.

Date  Event
9-06-2012 Kenji Fukaya (Simons Center)
Title: Gluing analysis and exponential decay estimate for pseudo-holomorphic curve with bubbles
9-13-2012 J. Elisenda Grigsby (Boston College)
Title: Categorified invariants and braid conjugacy

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Abstract: An “old” construction of Khovanov-Seidel associates to every braid a (homotopy equivalence class of) dg bimodules over an algebra. Their braid invariant is “faithful”–i.e., agrees on two braids iff the braids are the same. In this talk, I will describe a relationship between the Khovanov-Seidel braid invariant and the “sutured annular Khovanov homology” of the braid closure in the solid torus. I will also mention what this and some other categorified invariants can and cannot tell us about braid conjugacy classes. Parts of this talk describe joint work with D. Auroux and S. Wehrli, and other parts describe joint work with J. Baldwin.
9-20-2012 Leonid Polterovich (Tel Aviv)
Title: Symplectic topology of partitions of unity

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Abstract: We discuss quantitative aspects of the following phenomenon: certain finite open covers of closed symplectic manifolds do not admit Poisson commuting partitions of unity. The discussion goes in the language of Poisson bracket invariants whose definition involves elementary calculus, but whose study requires methods of “hard” symplectic topology.
9-27-2012 Joshua Batson (MIT) 10:15am
Title: Surfaces in 4-space and d-invariants

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Abstract: This talk provides background for my 1 p.m. talk on the nonorientable four-ball genus. First, I’ll give some background on the topology of surfaces in four-space–how to visualize them and their normal bundles, and the Gordon-Litherland construction of the knot signature. Then I’ll talk about gradings in Heegaard-Floer homology (an invariant of 3-manifolds and cobordisms between them). Those gradings can be used to define the d-invariants, which will be the key technical ingredient in my second talk.

Title: Nonorientable four-ball genus can be arbitrarily large

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Abstract: The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to 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.
10-04-2012 Zsuzsanna Dancso
Title: Odd Khovanov homology via hyperplane arrangements

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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.
10-11-2012  John Baldwin (Boston College)

Title: Szabo’s geometric spectral sequence for tangles

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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: A bordered monopole Floer theory

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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.
11-01-2012 Cancelled
11-08-2012 Moon Duchin (Tufts)
Title: Divergence of geodesics

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Abstract: I’ll define the divergence of geodesics and higher-dimensional analogs, emphasizing examples.  This gives an interesting family of geometric invariants “at infinity.”  I’ll discuss results on divergence in settings of interest for geometric topologists and geometric group theorists:  mapping class groups, Teichm├╝ller space, and right-angled Artin groups.
11-15-2012: David Shea Vela-Vick (Louisiana State)
Title:  The equivalence of transverse link invariants in knot Floer homology

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Abstract:  The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces.  Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined.  The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams.  The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo.  We show that these two previously defined invariant agree.  Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.
12-13-2012: Liam Watson (UCLA)
Title: Heegaard Floer homology solid tori

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Abstract: This talk will consider manifolds with simple bordered Heegaard Floer invariants. Focusing on the case of manifolds with torus boundary we will present, by analogy with Heegaard Floer homology lens spaces, a family of Heegaard Floer homology solid tori. These manifolds satisfy a version of the Alexander trick at the level of Heegaard Floer homology. The main goals of the talk will be to make these statements precise and put the examples in context.

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