SEMINAR ON SYMPLECTIC AND CONTACT GEOMETRY
Université Libre de Bruxelles
Second Semester 2009-2010
Monday 14:00-15:00
Room 2.NO.906
Tuesday, January 12th
Dishant Pancholi (ICTP, Trieste)
On possible generalizations of the Lutz twist
We discuss a few possibilities of generalizing the Lutz twist to higher dimension. One possible generalization allows us to construct an overtwisted contact structure in the given homotopy class (as plane fields) of a contact structure. This can be regarded as a possible generalization of the full Lutz twist. After discussing this construction in some detail we will discuss its limitations, other possible generalizations and related questions.
Monday, February 8th
Otto van Koert (Seoul National University)
Linearized Contact Homology of Connected Sums
Contact homology is an invariant of contact manifolds defined by counting
holomorphic curves in a symplectization. I will give an overview of several
versions of contact homology and discuss joint work with Frédéric
Bourgeois on connected sums. For connected sums of contact manifolds one can
show that many holomorphic curves cannot exist. This can then be used to show
that there is a long exact sequence for the linearized contact homology of
connected sums of contact manifolds.
| 14:15 |
Robert Vandervorst (Amsterdam) |
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Braid Floer homology |
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Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of the Floer homology originally used for the Arnol'd Conjecture yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding. Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear Cauchy-Riemann equations with respect to algebraic lengths of braids; (2) establishment of the topological invariance of the resulting braid Floer homology; (3) a shift theorem describing the effect of twisting braids in terms of shifting the braid Floer homology; (4) computation of examples; and (5) a forcing theorem for the dynamics of Hamiltonian disc maps based on braid Floer homology.
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| 15:45 |
Jack Waldron (Cambridge) |
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Knot theory and Fukaya categories |
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Symplectic Khovanov homology is an invariant of knots formulated in terms of Lagrangian Floer cohomology. It is conjecturally related to the Jones polynomial, for which it aims to give an intrinsically geometric setting. I shall describe how symplectic Khovanov homology fits into a TQFT described by a family of Fukaya categories. This is a first step towards proving the conjectured relation with the Jones polynomial. |
Monday, March 22nd
Andreas Ott (Zurich)
Gauged Gromov-Witten invariants via perturbation of the symplectic vortex equations
Gauged Gromov-Witten invariants are the gauge-theoretic generalization of Gromov-Witten invariants for symplectic manifolds equipped with a Hamiltonian Lie group action. These invariants are defined by counting solutions of the symplectic vortex equations. They were introduced by Cieliebak, Gaio, and Salamon for actions of arbitrary compact Lie groups on aspherical manifolds (i.e. where the symplectic form vanishes on all spherical homology classes) and by Mundet for semi-free circle actions on compact monotone manifolds. The main reason for the additional assumptions are complications in obtaining transversality for the boundary strata of the compactified moduli space of solutions of the vortex equations occurring in the presence of a group action. In this talk, I will present a perturbation scheme for the vortex equations that solves these transversality problems in a natural way, and explain how to define the gauged Gromov-Witten invariants for actions of arbitrary compact Lie groups on monotone symplectic manifolds.
| 14:15 |
Federica Pasquotto (VU Amsterdam) |
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Desingularisation of orbifolds obtained from symplectic reduction |
| 15:45 |
Jan Wehrheim (TU München) |
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Symplectic vortex equations and Givental's toric map space |
Thursday, April 22nd
Salle Debever
Daniel Mathews (Nantes)
Sutured Floer homology and contact-topological quantum field theory
We consider the topological quantum field theory properties of sutured
Floer homology, as introduced by Honda-Kazez-Matic. We present
several results in the "dimensionally reduced" case of product
manifolds. The SFH of such manifolds reduces to that of solid tori,
and forms a "categorification of Pascal's triangle". Contact
structures correspond to chord diagrams, and contact elements form
distinguished subsets of SFH of order given by the Catalan numbers. We
find natural "creation and annihilation operators" which allow us to
define a QFT-type basis of SFH, consisting of contact elements. In
fact sutured Floer homology in this case reduces to the combinatorics
of chord diagrams, and in a sense which can be made precise, is the
"quantum field theory of two non-commuting particles". The details of
this description have intrinsic contact-topological meaning, allowing
us for instance to compute certain contact categories, and to give a
"contact geometry free" proof that the contact element of a contact
structure with torsion is zero.
Monday, April 26th
Sheila Sandon (Nantes)
Equivariant generating functions and orderability of Lens spaces
In this talk I will first discuss (following Eliashberg, Kim and
Polterovich) the orderability problem for contact manifolds and its
relation to contact squeezing and non-squeezing phenomena. I will then
present an equivariant contact non-squeezing theorem for domains in R2n ×
S1 and explain (following Milin) how this result implies orderability of
Lens spaces. The proof of the equivariant contact non-squeezing theorem
will be based on an equivariant version of the contact homology groups for
domains in R2n × S1 constructed using generating functions.
| 14:15 |
Klaus Niederkrüger (Toulouse) |
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Weak fillings and holomorphic curves |
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I will explain how the classical non-fillability proof for overtwisted manifolds can be adapted to contact manifolds with positive Giroux torsion to show several (known) non-fillability results. And the end of the talk I will briefly sketch how related methods can be used to extract informations about planar contact manifolds. This is joint work with Chris Wendl.
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| 15:45 |
Viktor Fromm (Durham) |
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Circle actions and Morse homotopy
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Monday, May 17th
Julien Chenal (Nancy)
Géométrie de drapeaux généralisée et
structure multi-contact
O. Loos a montré que l'objet géométrique associé
à une algèbre de Lie Z/2Z-graduée est
un espace symétrique G/H. Ensuite, W. Bertram et K. H. Neeb ont
défini les géométries projectives
généralisées qui correspondent aux algèbres de
Lie 3-graduées. Dans cet exposé, on va définir un objet
géométrique associé à une algèbre de Lie
(2k + 1)-graduée, que j'appelle une géométrie de drapeaux
généralisée : (X+ = G/P-;
X- = G/P+) puis regarder la différence principale
entre le cas des 3-graduations et le cas général, k > 1. Cette
différence réside dans le fait que dans le cas
général, il existe sur les espaces X+ et
X- des structures (multi-)contact i.e une distribution de
filtrations sur X+ et X-.
Monday, May 31st
Till Broennle (Imperial College, London)
Deformation constructions of extremal Kähler metrics
I shall talk about constructing extremal Kahler metrics on the projectivisation of an unstable vector bundle. This provides us with new examples of the important class of extremal Kahler metrics. The main technique employed to prove existence results is an adiabatic limit, as already used in earlier works by Hong and Fine. I will talk about the general ideas and an outline of the existence proof, without worrying too much about the analytical details. Moreover, I shall point out how this technique could be extended to more general symplectic fibrations. Time permitting, I might explain more about the whole context of stability, extremal metrics and their deformations.
Previous Semesters :
First Semester 2005-2006
Second Semester 2005-2006
First Semester 2006-2007
Second Semester 2006-2007
First Semester 2007-2008
Second Semester 2007-2008
First Semester 2008-2009
Second Semester 2008-2009
First Semester 2009-2010
Next Semester :
First Semester 2010-2011
Maintained by
Frédéric Bourgeois.
