The SGGTC seminar meets on Fridays in Math 520, at 10:45 am unless noted otherwise (in red).
Michael Huthcings, "New symplectic embedding obstructions in four dimensions"
Abstract: After reviewing some background on symplectic embedding problems, I will describe some recently discovered obstructions to symplectically embedding one symplectic four-manifold (possibly disconnected, usually with boundary) into another. The obstructions are defined using embedded contact homology (ECH) of contact three-manifolds. For some interesting symplectic embedding problems which have been solved by Biran and McDuff, the ECH obstructions turn out to be sharp.
Katrin Wehrheim, "The PSS morphism for general symplectic manifolds"
Abstract: In joint work with Peter Albers and Joel Fish we study the [Piunikhin-Salamon-Schwarz]-map from Morse homology to Floer homology. For semi-positive symplectic manifolds it is given by counting pseudoholomorphic spheres with one cylindrical end and one marked point evaluating to a Morse trajectory space, and its inverse is given by reversing direction on the Floer end and Morse trajectory. We extend these definitions to morphisms PSS:HM-->HF and SSP: HF-->HM on general compact symplectic manifolds, using abstract transversality results in a polyfold description of the moduli spaces. Cobordism and grading arguments then prove that the composition of SSP with PSS is an isomorphism on Morse homology with Novikov coefficients; while there is no such cobordism for the reversed composition. However, this suffices to reprove the Arnold conjecture, bounding the number of Hamiltonian orbits below by the total rank of homology; which was previously proven by virtual moduli cycle techniques. Moreover, this could be combined with a spectral sequence argument to see that PSS is indeed an isomorphism.
Alexandru Oancea, "S^1-equivariant symplectic homology and linearized contact homology"
Abstract: The boundary part of S^1-equivariant symplectic homology of a Liouville domain is isomorphic, using rational coefficients, to linearized contact homology of the boundary. Applications on the contact homology side include a rigorous definition that solves transversality issues, a subcritical surgery long exact sequence, and the computation of linearized contact homology for unit cosphere bundles. On the symplectic homology side, applications include the definition of new algebraic operations. Joint work with F. Bourgeois.
Paul Kirk, "Geography of symplectic 4-manifolds with prescribed fundamental group"
Abstract: I'll discuss results on constructing closed 4 manifolds with prescribed > fundamental group efficiently, especially the case of symplectic 4-manifolds.
Emmanuel Ophstein, "Effective symplectic embeddings"
Abstract: I will explain the role played by Liouville forms and symplectic polarizations in the problems of embeddings. One of the aim is to give an effective version of Lalonde-McDuff's inflation method in the context of symplectic embeddings. I will try to discuss as many applications as I can among : full packings of symplectic manifolds by ellipsoids, or more general shapes, explicit maximal ball packings of P^2, or natural neighbourhoods of symplectic curves (in dimension 4).
Vera Vertesi, "Transverse positive braid satellites"
Abstract: Classification of transverse knots has been long investigated, and several invariants were defined for their distinction. One classical invariant is the self-linking number of the transverse knot, that can be given as the linking of the knot with its push off by a vectorfield in the contact planes that has a nonzero extension over a Seifert surface. Smooth knot types whose transverse representatives are classified by this classical invariant are called transversaly simple. In this talk I will talk I will prove transverse simplicity is inherited for positive braid satelites of some smooth knot types.
Shelly Harvey, "Combinatorial Spatial Graph Floer Homology"
Abstract: A spatial graph is an embedding, f, of a graph G into S^3. For each flat, balanced and oriented spatial graph, f(G), we define a combinatorial invariant HFG(f(G)) which is a bi-graded module over a polynomial ring in E +V variables, where V is the number of vertices and E is the number of edges in the graph. This invariant is a generalization of combinatorial link Floer homology defined by Manolescu, Ozsvath, Sarkar (MOS) for links in S^3. To do this, we define a grid diagram for a spatial graph and show that every embedding can be put into grid form. Following MOS, our invariant is the homology of a chain complex that counts certain rectangles in the grid. Although the chain complex depends on the choice of grid, the homology depends only on the embedding. This is joint work with Danielle O'Donnol (Smith College).
Eli Grigsby and Stephan Wehrli, "A relationship between Khovanov-type and Heegaard-Floer-type braid invariants I and II"
Abstract: Given a braid, one can associate to it a sequence of "categorified" braid invariants (one for each integer in a finite range) in
two apparently different ways: "algebraically," via the higher representation theory of U_q(sl_2) (using work of Khovanov-Seidel, Chen-Khovanov, and
Brundan-Stroppel), and "geometrically," using the bordered Floer invariants of its double-branched cover (defined by Lipshitz-Ozsvath-Thurston and
reinterpreted by Auroux). In this two-part talk, we will describe what we know so far about the connection between these invariants, focusing on
the relationship between the representation theory and the Floer theory.
The first part of the talk should be accessible to a "general" audience of topologists, and the second part will give more details about the constructions and the proofs. This is joint work with Denis Auroux.
Igor Kriz, "Field theories, homotopy realizations and Khovanov homology"
Abstract: I will talk about joint work with Po Hu and Daniel Kriz on realizations of 1+1-field theories valued in categories in modules over rigid ring spectra in stable homotopy theory. As an application, I will discuss an alternate approach to a recent result of Robert Lipshitz and Sucharit Sarkar constructing a stable homotopy refinement of Khovanov homology, and some related topics.
Eduardo Gonzalez, "Seidel Elements and Mirror Transformations"
Abstract: Let $X$ be a non-singular projective toric variety whose anti-canonical class is semipositive (nef). I will present work with Hiroshi Iritani regarding the relation of Givental's mirror symmetry transformations with Seidel's invertible elements in the Quantum Cohomology of $X$. If time permits I will describe a conjecture due to Chan-Lau-Leung-Tseng that relate this work to lagrangian Floer theory superpotentials introduced by Fukaya-Ohta-Ono-Oh.
Ciprian Manolescu, "The Heegaard Floer invariant of the circle"
Abstract: As part of the bordered Floer homology package, Lipshitz, Ozsvath and D. Thurston have associated to a parametrized oriented surface a certain differential graded algebra. I will describe a decomposition theorem for this algebra, corresponding to cutting the surface along a circle. In this decomposition, we associate to the circle a categorical structure called the nilCoxeter sequential 2-algebra. I will also discuss a decomposition theorem for bordered modules associated to nice diagrams, corresponding to cutting a 3-manifold with boundary along a surface transverse to the boundary. This is joint work with Christopher Douglas.
Garrett Alston, "Real Lagrangians in the quintic"
Abstract: I will talk about some computations of Floer cohomology of real Lagrangians in the quintic, and I'll relate these computations to matrix factorizations in the mirror of the quintic. I'll also make some remarks about the possibility of the real Lagrangians generating the Fukaya category. This is work in progress.
Jo Nelson, "Cylindrical contact homology as a well-defined homology theory?"
Abstract: Symplectic field theory has been around for more than a decade but significant analytic obstacles remain and very few rigorous proofs have appeared in the literature. Cylindrical contact homology is one of the "simplest" invariants coming from this framework, but even the full details of its construction have not been lovingly worked out. In fact it has recently come to light that the usual assumptions on the Conley-Zehnder indices of contractible closed Reeb orbits do not ensure d^2=0 due to the existence of multiply covered cylinders and their branched covers. In this talk I will explain these issues with concrete examples and explore what stronger conditions are necessary on the growth rates of the indices of simple contractible orbits to obtain a homological invariant. I will also sketch a method in progress that seems to avoid these issues for prequantization spaces and certain S^1 bundles over nicely behaved symplectic orbifolds.
Cotton Seed, "Twisting Szabo's geometric spectral sequence"
Abstract: Recently, by studying suitably twisted complexes, a number of knot homology theories have been formulated in terms of complexes generated by the spanning trees of a knot. In this talk, I will describe a twisted version of Szabo's geometric spectral sequence in Khovanov homology. To begin, I will review related constructions: Roberts' totally twisted Khovanov homology, a twisted variant of the spectral sequence from Khovanov homology to the double-branched cover, and Szabo's geometric spectral sequence. I will present my construction and give some computational results. Finally, I will describe some natural directions for future work.
Andras Stipsicz, "Knots in Lattice homology"
Abstract: In 2008 Nemethi introduced an invariant of negative definite plumbings, called lattice homology. We introduce a filtration on the lattice homology of a negative definite plumbing tree associated to a further vertex and show how to determine lattice homologies of surgeries on this last vertex. We discuss the relation with Heegaard Floer homology.
Other relevant information
Interactions Between Algebra and Dynamics in Symplectic Topology
June 17 – 21, 2012
CAST Summer School and Conference
Alfred Renyi Institute
July 9 – 20, 2012
- Workshop on Symplectic Field Theory VI
July 23 – 27, 2012
- Workshop and Conference on Holomorphic Curves and Low Dimensional Topology
Jul 30 – Aug 11, 2012
- Columbia Geometric Topology Seminar
- Columbia Algebraic Geometry Seminar
- Eilenberg lecture series
- Princeton Topology Seminar
Announcements for this seminar, as well as for related seminars and events, are sent to the "Floer Homology" e-mail list maintained via Google Groups. You can subscribe directly via Google Groups or by contacting R. Lipshitz.