Pdf file

Title: ECH=HF via open book decompositions I

Title: Symplectic homology of 4-dimensional Weinstein manifolds and Legendrian homology of links

Abstract: We show how to compute the symplectic homology of a 4-dimensional Weinstein manifold from a diagram of the Legendrian link which is the attaching locus of its 2-handles. The computation uses a combination of a generalization of Chekanov's description of the Legendrian homology of links in standard contact 3-space, where the ambient contact manifold is replaced by a connected sum of S^2 \times S^1's, and resent results on the behavior of holomorphic curve invariants under Legendrian surgery.

Title: Contact geometry, open books and monodromy

Abstract: Recall that an open book decomposition of a 3-manifold M is a link L in M whose complement fibers over the circle with fiber a Seifert surface for L. Giroux's correspondence relates open book decompositions of a manifold M to contact structures on M. This correspondence has been fundamental to our understanding of contact geometry. An intriguing question raised by this correspondence is how geometric properties of a contact structure are reflected in the monodromy map describing the open book decomposition. In this talk I will show that there are several interesting monoids in the mapping class group that are related to various properties of a contact structure (like being Stein fillable, weakly fillable, . . .). I will also show that there are open book decompositions of Stein fillable contact structures whose monodromy cannot be factored as a product of positive Dehn twists. If time permits I will also discuss how natural constructions on one side of the Giroux correspondence affect the other. This is joint work with Jeremy Van Horn-Morris and Ken Baker.

Title: Spectral invariant with bulk and its applications

Abstract: Spectral invariant of Floer homology of Hamiltonian system is studied by Viterbo, Schwartz, Oh and etc. It is applied to the study of group of Hamiltonian diffeomorphisms by Entov-Polterovich etc.
I will explain its variant related to the bulk deformation of quantum cohomology. (big quantum cohomology.)
We can use it together with its relation to Lagrangian Floer theory to show some interesting phenomenon on the group of Hamiltonian diffeomorphisms.
This is a joint work with Oh, Ohta, Ono.

Title: An invariant for knots in a contact manifold

Abstract: Given a contact structure on a 3-manifold, Y, I will define an integer-valued invariant of knots in Y by combining the knot Floer homology invariants and the contact invariant in Ozsváth-Szabó theory. Applications will be discussed.

Title: The chord conjecture in three dimensions

Abstract: We prove that every Legendrian knot in a closed oriented 3-manifold with a contact form has a Reeb chord. The proof uses cobordism maps on embedded contact homology defined via Seiberg-Witten theory. This is a joint work with Cliff Taubes.

Title: Moduli space of holomorphic curves relative singular divisors

Title: On the compatible contact structures of fibered Seifert links in homology 3-spheres

Abstract: The notion of compatible contact structures of open book decompositions of 3-manifolds was introduced by W. Thurston and H. Winkelnkemper, and developed by E. Giroux. We are studying which open book decompositions are compatible with tight contact structures. The answer to this question is very simple for the fibered Seifert links in positively-twisted Seifert fibered homology 3-spheres.

Theorem: Let L be a fibered Seifert link in a positively-twisted Seifert fibered homology 3-sphere. Then, the compatible contact structure of L is tight if and only if the orientation of L is consistent with the orientation of fibers of the Seifert fibration.

To prove this theorem, we construct the compatible contact structure explicitly. In this talk, we present this construction and also explain some further studies.

Title: Cobordisms of sutured manifolds

Abstract: Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology induced by decorated knot cobordisms.

Title: Instantons, Chern-Simons invariants, and Whitehead doubles of (2, 2^k-1) torus knots

Abstract: We revisit an argument of Furuta, using SO(3) instanton moduli spaces on 4-manifolds with boundary and estimates of Chern- Simons invariants of flat SO(3) connections on 3-manifolds to prove that the infinite family of untwisted positive clasped Whitehead doubles of the (2, 2^k-1) torus knots are linearly independent in the smooth knot concordance group. (joint work with Matt Hedden)

Title: ECH=HF via open book decompositions II

Title: Deformations of circle-valued Morse functions on surfaces

Abstract: Let X be a compact orientable surface with boundary and M(X,S^1) be the space of all Morse maps X --> S^1 whose restrictions to the boundary of X are covering maps. The aim of this talk is to describe the path components of the space M(X,S^1). This description extends the results of S.Matveev, V.Sharko, and the author about path components of the space of Morse mappings being locally constant on the boundary of X.

Title: Enhanced knot contact homology and transverse knots

Abstract: We describe an enhanced version of knot contact homology that yields invariants of topological knots and transverse knots in R^3. In particular, this constitutes a surprisingly effective invariant of transverse knots, and we are able to show transverse nonsimplicity for several new knot types.

Title: Lagrangian Floer theory on compact toric manifolds

Abstract: K. Fukaya, Y.-G. Oh, H. Ohta and I have developed Floer theory of Lagrangian submanifolds. In the case of Lagrangian torus fibers in compact toric manifolds, the theory is governed by the potential function. I plan to explain some application of this theory.

Title: Homology cylinders and knot theory

Abstract: This is a joint work with Hiroshi Goda (Tokyo University of Agriculture and Technology).
Homology cylinders over a surface defined by Goussarov and Habiro play an important role in the recent theory of mapping class groups and finite-type invariants of 3-manifolds. Several invariants have been constructed to study the structure of the monoid and related groups of homology cylinders.
In this talk, we focus on higher-order Alexander invariants of homology cylinders and discuss how they can be applied to knot theory by introducing a class of knots called homologically fibered knots. Relationships to (the decategorification of) sutured Floer homology are also mentioned.

Title: Seiberg-Witten invariants and end-periodic Dirac operators

Abstract: This research is a part of the ongoing project with Tomasz Mrowka and Daniel Ruberman. We study the Seiberg-Witten equations on smooth spin 4-manifolds X with integral homology of S^1 \times S^3. The count of their solutions, known as the Seiberg-Witten invariant of X, depends on the choices of Riemannian metric and perturbation. We resolve this dependency issue by introducing a correction term which is in essence the L^2 index of the Dirac operator on a manifold with periodic end modeled on the infinite cyclic cover of X. The corrected count is a smooth invariant of X whose reduction is the Rohlin invariant. We discuss some calculations of this invariant, as well as our progress towards the general index theorem on manifolds with periodic ends.

Title: Tight contact structures and Dehn surgery

Abstract: Suppose that the 3-manifold Y is given by Dehn surgery on a knot in the 3-sphere S^3. We show conditions under which Y admits tight contact structures. We apply Heegaard Floer theory in proving tightness.

Title: Bordered Floer Homology

Abstract: Heegaard Floer homology has the structure of a 4-dimensional TQFT, assigning chain complexes to 3-manifolds and maps between them to 4-dimensional cobordism between 3-manifolds. We extend this theory down one step further, assigning differential algebras to surfaces and differential bimodules to a 3-dimensional cobordisms between surfaces. This lets us give, for instance, another algorithm for computing Heegaard Floer homology of an arbitrary 3-manifold.
This is joint work with Robert Lipshitz and Peter Ozsváth.

Title: Legendrian and Transverse Classification of twist knots

Abstract: In 1997 Chekanov gave the first example of a knot type whose Legendrian representations are not distinguishable using only the classical invariants: the $5_2$ knot. Epstein, Fuchs and Meyer extended his result by showing that there are at least $n$ different Legendrian representations of the $(2n+1)_2$ knot with maximal Thurston-Bennequin number. The aim of this talk to give a complete classification of Legendrian representations of twist knots. In particular the $(2n+1)_2$knot has exactly $\lceil \frac{n^2}{2} \rceil$ Legendrian representations with maximal Thurston-Bennequin number. This is a joint work with John Etnyre and Lenhard Ng.