20032004 

43eme édition 

21 juin  2 juillet 2004 
Organisateurs : Octav Cornea
(Montréal) et Paul Biran (Tel Aviv) 
The talks at this meeting centered on Morse theoretical
techniques which can be used to solve difficult analytic
problems as well as problems in symplectic topology and in
robotics.The key tool in modern symplectic topology is Floer
homology and related techniques and this constituted a
recurring theme for the lectures given. Some of the topics
discussed (in particular by Hofer, Schwarz, Cornea and
Polterovich) were presented publicly for the first time at the
ASI. As well many of the talks of different speakers were
strongly interrelated which contributed to the overall quality
and strength of the ASI itself. 
Alberto ABBONDANDOLO (Scuola Normale
Superiore) 
The Morse complex for infinite dimensional
manifolds 
Paul BIRAN (TelAviv University) 
Lagrangian Geometry and Topology Morse
theory 
Ralph COHEN (Stanford University) 
Morse theory, Graphs, and Loop spaces 
Octav CORNEA (Université de
Montréal) 
Homotopical methods in MorseFloer theory 
Michael FARBER (TelAviv University) 
Topology of robot motion planning

Kenji FUKAYA (University of Kyoto) 
Floer homology of Lagrangian submanifolds 
Helmut HOFER (Courant Institute) 
The analytic background of Symplectic Field
Theory 
Marek ISYDOREK (Technical University
of Gdansk) 
The Conley index in Hilbert spaces with
applications 
YongGeun OH (University of
Wisconsin at Madison) 
Chain level Floer theory and geometry of
Hamiltonian diffeomorphism group 
Leonid POLTEROVICH (TelAviv
University) 
Floer homology and symplectic dynamics 
Matthias SCHWARZ (University of
Leipzig) 
Lectures on Floerhomological methods in
symplectic geometry 
Claude VITERBO (École Polytechnique
 Palaiseau) 
Generating functions and applications 
In the first week Helmut Hofer talked
about the foundations of symplectic field theory  this is one of the major new "machines" in symplectic topology whose development
has been pursued by Hofer, Eliashberg and Givental for a number of
years now. The origins of symplectic field theory lie in Floer's
machinery but it goes much beyond that both in applications as well
as in complexity. Hofer's lectures were therefore extremely timely.
Matthias Schwarz presented his recent proof for a result of Viterbo
relating the Floer homology of the cotangent bundle and the string
topology of the zero section of this bundle. This notion of string
topology has been recently introduced by Chas and Sullivan with a
purely topological motivation and the fact that this notion fits
perfectly with the quantum product in Floer theory is quite
remakable. 
Michael Farber discussed applications to
robotics. Paul Biran described efficient methods to use Floer
homology in the monotone case to prove results concerning the
topological structure of Lagrangian submanifolds. Leonid Polterovich
showed how to relate this symplectic topology to dynamics and
geometric group theory methods. Octav Cornea presented higher order
Floer type invariants and applications.
This was an intense first week with lectures of the highest order of
interest for specialists as well as for beginners in the field. The
second week continued as strongly: 
Claude Viterbo talked about generating
functions techniques, Alberto Abbondandolo discussed Morse theory in
Hilbert spaces, Kenji Fukaya talked about a new version of his $A^{\infty}$
machinery (developed with Oh and Ono) and applications, some of
which overlapped with applications obtained by different methods by
Cornea jointly with Lalonde and mentioned in the first week. Marek
Izydorek lectured on his approach to the infinite dimensional Conley
index and YongGeun Oh presented his recent spectral invariant
techniques and chain level Floer methods. Finally, Ralph Cohen
described his topological approach to string topology and
itspotential implications for symplectic topology. 
