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General information:
The workshop will focus on open problems in contact and symplectic topology. It will be held on September 12-15, 2011 in
Lodz, Poland,
as a special section of the
Joint Meeting of the Israel Mathematical Union and the Polish Mathematical Society
and is
sponsored by
CAST - Contact and Symplectic Topology
(a research network supported by the European Science Foundation).
Program:
Schedule:
Monday, September 12
14:30-15:20 Usher
15:30-16:00 Shelukhin
16:00-16:30 COFFEE BREAK
16:30-17:20 Welschinger
17:30-18:20 Abouzaid
18:30-19:30 Fukaya
Tuesday, September 13
8:30 - 9:15 Niederkrueger
9:20 - 10:00 Gal
14:00-14:50 Ostrover
15:00-16:00 Lalonde
16:00-16:30 COFFEE BREAK
16:30-17:20 Hind
17:30-18:30 Abreu
Wednesday, September 14
8:30 - 9:00 Pedroza
9:10 - 10:00 Ono
Thursday, September 15
8:30-9:20 Kedra
14:00-14:50 Savelyev
15:00-15:50 Frauenfelder
Titles and abstracts:
- Mohammed Abouzaid (MIT, USA)
- Title: Generating Fukaya categories
- Abstract:
I will discuss a geometric context wherein one can prove that a
certain collection of Lagrangians generate the Fukaya category, and
develop some applications to the study of Lagrangian embeddings.
- Miguel Abreu (IST, Portugal)
- Title: Remarks on Lagrangian Intersections in Toric Manifolds
- Abstract:
I will show how to get a lot of "mileage" out of applying two
simple
geometric remarks to questions on Lagrangian intersections in toric
manifolds.
Joint work with Leonardo Macarini.
- Urs Frauenfelder (Seoul National U., South Korea)
- Title: Bubbles and Onis
- Abstract:
In joint work with Peter Albers we study a version of Rabinowitz action functional on very negative line bundles over a symplectic manifold. If the line bundle is very negative bubbles disappear. I explain how bubbles transform into oni's - a new PDE problem - and how one can fight against oni's . Then I explain how one can use this approach to attack
Arnold's conjecture in the presence of torsion.
- Kenji Fukaya (U. of Kyoto, Japan)
- Title: Deformation of surface singularity and Lagrangian Floer homology
- Abstract:
In this talk I explain calculation of Floer homology of
some Lagrangian tori in a symplectic manifold obtained
by deforming orbifold singularity in toric surface.
Especially I explain the example of a cubic surface.
The calculation implies that
the second bounded cohomology of
the universal cover of its group of Hamiltonian diffeomorphisms
is infinite-dimensional.
- Swiatosław Gal (U. Wien, Austria)
- Title: dK(g):=g*α-α
- Abstract:
Let G be an abstract group acting on a space X. Let g be an element of
G. It is a vague, but interesting problem what could be deduced about group
properties of g∈G from the dynamical properties of the action of g.
In the talk we would construct a natural cocycle on a class of groups of
homeomorphisms. This class includes, among others, groups of Hamiltonian
diffeomorphisms of symplectically aspherical manifolds and groups of
homeomorphisms of a space preserving given first cohomology class.
We will use this cocycle to prove that certain elements cannot be distorted
(they have nonzero translation length).
As a side remark we will deduce that fundamental group of symplectically
hyperbolic manifold cannot be amenable.
This is a joint work with Jarek Kędra from U. of Aberdeen.
- Richard Hind (U. of Notre Dame, USA)
- Title: Symplectic embedding of ellipsoids
- Abstract:
We will discuss some constructions and obstructions to embedding ellipsoids in dimension greater than four.
There are many open questions, but optimal results are known for certain classes of embeddings into
balls and cylinders. This is joint work with Olga Buse and Ely Kerman.
- Jarek Kędra (U. of Aberdeen, UK, and U. of Szczecin, Poland)
- Title: Quasi-isometric embeddings into diffeomorphism groups
- Abstract:
Let M be a smooth compact oriented manifold of dimension at least
two endowed with a volume form. Assuming certain conditions on the
fundamental group of M I will construct quasi-isometric embeddings of
either free abelian or direct products of non-abelian free groups into
the group of volume preserving diffeomorphisms of M equipped with
the L^p metric induced by a Riemannian metric on M.
This is a joint work with Michael Brandenbursky.
- François Lalonde (U. of Montreal, Canada)
- Title: Moduli spaces in cluster homology
- Klaus Niederkrueger (U. Paul Sabatier - Toulouse III, France)
- Title: Weak fillability for higher dimensional contact manifolds
- Abstract:
In $3$--dimensional contact topology there is a well-known
hierarchy of definitions of fillability, including
Stein, exact, strong, and weak symplectic fillings.
These notions have been studied extensively yielding examples
that show that they all differ.
In higher dimensional contact topology similar definitions
exist, but the proposed definition for weak fillings
has been known for at least 20 years to be equivalent
to the strong fillability.
In this talk, I will propose a new definition for weak
fillability, and try to justify it, by showing that
every contact structure can be modified into one that
is not weakly fillable (with our definition), and
that there exist contact manifolds that are weakly,
but not strongly fillable.
This is a joint work with Patrick Massot and Chris Wendl.
- Kaoru Ono (Hokkaido U., Japan)
- Title: Displacement energy of Lagrangian submanifolds and
the torsion of Lagrangian Floer cohomology
- Abstract:
Yuri Chekanov gave a lower bound for the displacement
energy of a Lagrangian submanifold using Floer theoretical argument.
In this talk, I will give a lower bound for the displacement energy
of an unobstructed Lagrangian submanifold using the torsion information
of Lagrangian Floer cohomology with $\Lambda_0$-coefficients.
As examples, we discuss the displacement energy of Lagrangian torus
fibers in compact toric manifolds. This talk is based on a joint work
with K. Fukaya, Y.-G. Oh, H. Ohta.
- Yaron Ostrover (Tel-Aviv U., Israel)
- Title: On the uniqueness of Hofer's geometry
- Abstract:
In this talk we address the question whether Hofer's metric
is unique among the Finsler-type bi-invariant metrics on the group of
Hamiltonian diffeomorphisms. The talk is based on a recent joint work
with Lev Buhovsky.
- Andrés Pedroza (U. de Colima, Mexico)
- Title: Bounded Hamiltonian diffeomorphisms
- Abstract:
We prove the bounded isometry conjecture of F.Lalonde and L.Polterovich
for a special class of closed symplectic manifolds. The class of symplectic
manifolds are those whose degree-one cohomology group is generated by
embedded punctured tori.
- Yasha Savelyev (CRM-Montreal, Canada)
- Title: Quantum corrections to Hofer length and diameter
- Abstract:
We discuss a Floer theoretic, variationally complete length functional
on the path spaces of Ham(M,\omega), inducing the spectral norm.
The main motivation for this is to give an abstract approach to
injectivity radius problem in Hofer geometry, and we will spend some
time discussing this.
- Egor Shelukhin (Tel-Aviv U., Israel)
- Title:
A quasimorphism for every symplectic manifold
- Abstract:
Using the action of the Hamiltonian group on the space of compatible almost complex structures we construct a quasimorphism on its universal cover.
Key notions in the construction include equivariant moment maps, the Action homomorphism and bounded cocycles given by integration over geodesic triangles.
- Michael Usher (U. of Georgia, USA)
- Title:
Hofer's metrics and the boundary depth
- Abstract:
I will show how to use a Floer-theoretic quantity called the boundary depth to prove infinite diameter (and indeed the existence of infinite-dimensional quasi-isometrically embedded normed vector spaces) for Hofer's metrics on groups of Hamiltonian diffeomorphisms or spaces of Lagrangian submanifolds in a variety of cases.
- Jean-Yves Welschinger (U. Lyon 1, France)
- Title:
What is the total Betti number of a random real hypersurface?
- Abstract:
I will estimate from above the expected total Betti number of a random real
hypersurface of a smooth real projective manifold. This upper bound is based
on the equirepartition of critical points of a real Lefschetz pencil restricted
to such a hypersurface. The proofs involve Ho"rmander's peak sections and
a formula of Poincare'-Martinelli, it is a joint work with Damien Gayet.
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