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 lowdimensional topology, while the methods developed in 3 and 4dimensional topology found applications to symplectic geometric problems.
For instance, the Lagrangian intersection theory, and in particular, Lagrangian Floer homology theory which was created for symplecticgeometric 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 3manifolds. 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 5dimensional 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 threemanifolds 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 SeibergWitten Floer homology for 3manifolds. A promising current development in the symplectic topology of highdimensional 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 
9062012  Kenji Fukaya (Simons Center) Title: Gluing analysis and exponential decay estimate for pseudoholomorphic curve with bubbles 
9132012  J. Elisenda Grigsby (Boston College) Title: Categorified invariants and braid conjugacy Homepage: https://www2.bc.edu/~grigsbyj/ show abscractAbstract: An “old” construction of KhovanovSeidel 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 KhovanovSeidel 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.

9202012  Leonid Polterovich (Tel Aviv) Title: Symplectic topology of partitions of unity show abstractAbstract: 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.

9272012  Joshua Batson (MIT) 10:15am Title: Surfaces in 4space and dinvariants show abstractAbstract: This talk provides background for my 1 p.m. talk on the nonorientable fourball genus. First, I’ll give some background on the topology of surfaces in fourspace–how to visualize them and their normal bundles, and the GordonLitherland construction of the knot signature. Then I’ll talk about gradings in HeegaardFloer homology (an invariant of 3manifolds and cobordisms between them). Those gradings can be used to define the dinvariants, which will be the key technical ingredient in my second talk.
1:15pm Title: Nonorientable fourball genus can be arbitrarily large show abstractAbstract: The nonorientable fourball 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 fourball genus, which can be bounded using algebraic topology, HeegaardFloer homology, and Khovanov homology, the best lower bound in the literature on the nonorientable fourball genus for any K is 3. We find a lower bound in terms of the signature of K and the HeegaardFloer dinvariant of the integer homology sphere given by 1 surgery on K. In particular, we prove that the nonorientable fourball genus of the torus knot T(2k,2k1) is k1.

10042012  Zsuzsanna Dancso Title: Odd Khovanov homology via hyperplane arrangements show abstractAbstract: We will describe a construction of Odd Khovanov homology (isomorphic to that of OzsvathRasmussenSzabo) 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.

10112012  John Baldwin (Boston College) Homepage: https://www2.bc.edu/john 10:15am
Title: Szabo’s geometric spectral sequence for tangles
show abstractAbstract: 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 “functorvalued invariant of tangles.” I’ll introduce Ainfinity 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 oncepunctured 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.
1:15pm
Title: A bordered monopole Floer theory
show abstractAbstract: I’ll discuss workinprogress toward constructing monopole Floer theoretic invariants of bordered 3manifolds. Roughly, our construction associates an Ainfinity algebra to a surface, an Ainfinity module to a bordered 3manifold, and a map of Ainfinity modules to a 4dimensional cobordism of bordered 3manifolds. 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 3manifolds with that of the manifold obtained by gluing the former together along homeomorphic components of their boundaries. This is joint work with Jon Bloom.

11012012  Cancelled 
11082012  Moon Duchin (Tufts) Title: Divergence of geodesics show abstractAbstract: I’ll define the divergence of geodesics and higherdimensional 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, Teichmüller space, and rightangled Artin groups.

11152012:  David Shea VelaVick (Louisiana State) Title: The equivalence of transverse link invariants in knot Floer homology Homepage: https://www.math.lsu.edu/~ show abstractAbstract: The Heegaard Floer package provides a robust tool for studying contact 3manifolds 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.

12132012:  Liam Watson (UCLA) Title: Heegaard Floer homology solid tori Homepage: http://www.math.ucla.edu/~ show abstractAbstract: 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.
