Elliptic Methods and Moduli Spaces (1257)
Méthodes elliptiques et espaces de modules

Dates: 21-25 September 2015 at CIRM (Marseille, France)


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 Fukaya-Oh-Ohta-Ono which is a contravariant theory, and the covariant Cornea-Lalonde 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 Biran-Cornea 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 3-manifolds, which has been introduced by Floer himself in connection with Donaldson theory, and is obtained using the Chern-Simons functional on the space of connections on a principal SU(2)-bundle. A similar Floer theory has been developed in connection with Seiberg-Witten (monopole) 4-dimensional invariants. In general, the Floer homology associated with a 4-dimensional gauge theory is the target of the corresponding relative invariant (defined for 4-manifolds with boundary). New important developments concern applications of these Floer theories and their relations with other homology theories for 3-manifolds: embedded contact homology, Heegaard-Floer 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.


Lagrangian Floer Cohomology in Families

Lagrangian Topology and Cobordisms

Structure and Applications of Symplectic Homology

Persistence Modules and Hamiltonian Diffeomorphisms

Topology of Random Nodal Sets