[geometry-ml:01060] Knots, Contact Geometry and Floer Homology

[Hiroshi Goda]

皆様

５月２４日ー２８日に東大数理にて下記研究集会を開催いたしますので，
ご案内いたします．

追加情報，変更などありましたら web site

http://faculty.ms.u-tokyo.ac.jp/~kch2010/index.html

にてお伝えしますので，ご留意下さい．


東京農工大学
合田洋




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             Knots, Contact Geometry and Floer Homology
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                 The University of Tokyo
                    May 24 - 28, 2010

Lecture Hall
Graduate School of Mathematical Sciences
The University of Tokyo

May 24 (Mon)
 9:30--10:30　Andras Juhasz  (University of Cambridge)
11:00--12:00　Paul Kirk (Indiana University)
13:30--14:30　Matt Hedden  (Michigan State University)
15:00--16:00　Dylan Thurston  (Columbia University)
16:20--17:20  Takuya Sakasai  (Tokyo Institute of Technology)

May 25 (Tue)
 9:30--10:30　Michael Hutchings  (University of California, Berkeley)
11:00--12:00　Sergiy Maksymenko (Institute of Mathematics of NAS of Ukraine)
13:30--14:30　Kaoru Ono (Hokkaido University)
15:00--16:00　Tobias Ekholm (Uppsala University)
16:20--17:20  Vera Vertesi (MSRI, Berkeley)
18:00--20:00  Banquet

May 26 (Wed)
 9:30--10:30　Nikolai Saveliev  (The University of Miami)
11:00--12:00　Kenji Fukaya  (Kyoto University)

May 27 (Thu)
 9:30--10:30　Andras Stipsicz  (Renyi Institute of Mathematics)
11:00--12:00　Lenhard  Ng  (Duke University)
13:30--14:30　Vincent Colin (University of Nantes)
15:00--16:00　John Etnyre  (Georgia Institute of Technology) 
16:20--17:20  Masaharu Ishikawa (Tohoku University)

May 28 (Fri)
 9:30--10:30  Eleny Ionel  (Stanford University)
11:00--12:00　Ko Honda  (University of Southern California)


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Titles and Abstracts:


Tobias Ekholm
 
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.

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John Etnyre
 
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.
 
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Matt Hedden
 
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 
Ozsvath-Szabo theory. Applications will be discussed.

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Masaharu Ishikawa
 
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.

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Andras Juhasz 
 
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.

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Paul Kirk 
 
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)

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Lenhard Ng 
 
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.

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Kaoru Ono
 
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.

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Takuya Sakasai
 
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.

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Nikolai Saveliev
 
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.

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Andras Stipsicz
 
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.

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Vera Vertesi
 
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.


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Organizing Committee : 
H. Goda, T. Kalman, T. Kohno, A. Pajitnov