DESCRIPTION
Nowadays, most of Symplectic topology and Gauge theory are based on a very diverse and profound set of Floer theories, which are themselves derived from a rich and complex corpus of moduli spaces. Here is a short list of these theories, that have all followed the Floer theory, but in contexts that are far more general or in contexts that seem a priori radically different:  The Floer theory itself on Lagrangian submanifolds that has led to two universal theories: the one by FukayaOhOhtaOno which is a contravariant theory, and the covariant CorneaLalonde Cluster theory developed more recently and independently.  The Embedded Contact Homology introduced by Hutchings, that has had several far reaching applications: the proof of the full Weinstein conjecture on Contact manifolds, the construction of invariants, that are formally capacities, associated to contact or symplectic manifolds, and that are sharp in many cases.  The Symplectic field theory introduced by Eliashberg, Givental and Hofer in 2000, that is supposed to be the ultimate field theory in physics, and on which there are current trends of research at the highest level, especially on moduli spaces associated to the theory, on compactness results and on regularity. This is exactly the kind of theory that prompted the birth of polyfolds which is still in progress and is supposed to be the ultimate theory of moduli spaces where elliptic PDE's intervene.  Cobordisms and rigidity of Lagrangian submanifolds: this is a theory developed by BiranCornea that shows that monotone Lagrangian submanifolds enjoy a form of strong rigidity, which is absent from general theories. For instance, the fact of preserving the monotonicity through Lagrangian surgeries imposes a hard constraint that makes the whole monotone cobordism theory appealing, with many applications.  The first Floer homology was the instanton Floer homology for 3manifolds, which has been introduced by Floer himself in connection with Donaldson theory, and is obtained using the ChernSimons functional on the space of connections on a principal SU(2)bundle. A similar Floer theory has been developed in connection with SeibergWitten (monopole) 4dimensional invariants. In general, the Floer homology associated with a 4dimensional gauge theory is the target of the corresponding relative invariant (defined for 4manifolds with boundary). New important developments concern applications of these Floer theories and their relations with other homology theories for 3manifolds: embedded contact homology, HeegaardFloer homology and Khovanov homology. In order to present the basic elements of these theories, the best experts were invited to give courses during the 1 week workshop. 
SCIENTIFIC & ORGANIZING COMMITTEE
SPEAKERS

CIRM  JeanMorlet Chair
François Lalonde & Andrei Teleman
Moduli Spaces in Symplectic Topology and Gauge Theory
Espaces de modules en topologie symplectique et théorie de Jauge