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Moduli Spaces in Symplectic Topology and in Gauge Theory (1256)
Espaces de modules en topologie symplectique et théorie de Jauge

Dates: 1-5 June 2015 at CIRM (Marseille, France)

Symplectic topology can be considered as the mathematical versant of String theory. They were discovered independently at the same time in the 80’s. The second one is a fantastic enterprise to unify low-scale and highscale physics, while the first one was born as a tool to resolve the extraordinarily difficult problems of closed orbits in non-integrable generic Hamiltonian systems (the famous Arnold and Weinstein conjectures). Since that time, both theories have developed into a far reaching mathematical endeavour and much of today’s attention from the geometers across the world is directed towards the many conjectures of Symplectic Topology. Symplectic Topology is, together with Number Theory, the only field that seems able to produce very simple conjectures that are notoriously hard to prove. It is also the only theory, to our knowledge, that produces deep and rich moduli spaces at such a pace! This workshop will bring together the best specialists in the world around the problems of moduli spaces in Symplectic topology and Gauge theory. These rich moduli spaces are always set up to define functors or morphisms depending on pertinent non-linear elliptic PDE’s configurations, often coupled with trees of Morse flows. Our understanding of these moduli spaces is based on (1) the appropriate setting for these moduli spaces to get the right compactification needed (of which Uhlenbeck’s and the Gromov’s compactification theorems are just the very first basic blocks), and (2) the construction of the algebraic structures that prevail in these moduli spaces and that, ultimately, govern the whole Floer-SFT-like theory. So the workshop was divided along these lines in the following way:
1. Analytic foundations and applications to dynamics.
That part of the workshop focused on the following three subjects that are in fast development:
    a. Analytical foundations of Symplectic Field Theory. The main development in this direction is that the monumental work of Hofer-Wisocki-Zehnder is now reaching cruising speed and is now starting to be understood and applied by more and more researchers. In particular, it is expected that in a few years the foundations of the various symplectic field theories will become solid.
    b. Closed orbits of Hamiltonian flows, symplectic dynamics and Seiberg-Witten Floer homology. A number of spectacular results have been obtained recently in this direction. For instance, Taubes’ proof of the Weinstein conjecture is related to the embedded contact homology of Hutchings, and extensions of his proof of this conjecture yielded the famous isomorphism between embedded contact homology and Seiberg-Witten Floer homology.
Another application of the theory is Ginzburg’s proof of the Conley conjecture. In a different direction, we mentioned the dynamical perspective on the study of groups of Hamiltonian   diffeomorphisms provided by the « quasi-morphisms » work of Entov-Polterovich.
    c. Mean curvature flows for Lagrangian submanifolds. This is a direction that only is starting to get on the « screen » these days but will become quite signicant in the years to come. It is concerned with properties of the mean curvature flows applied to Lagrangian submanifolds as discussed in the work of Yau, Joyce, Smoczyk, Schwarz, Neves, Tian and others.

2. Algebraic structures and ramifications.
There are three subjects in this direction that were discussed:
    a. Further algebraic structures. The complexity of algebraic structures used today in symplectic topology is quite high but even more sophisticated constructions are attempted these days by various authors, especially Fukaya et al., Eliashberg and collaborators, Seidel, Abouzaid, Auroux in particular. This is sometimes done in relation to Mirror Symmetry (Seidel, Abouzaid, Auroux ) or in relation to Lagrangian topology (for instance by Cornea-Lalonde, Biran-Cornea, Hu-Lalonde).
    b. Enumerative invariants for Lagrangian submanifolds. A topic of much interest these days, these constructions are reflected in work on « real » symplectic topology as pursued by Welschinger, Solomon and others. There are also other developments in the Calabi-Yau case (by Yau, Fukaya, Iacovino) as well as in the monotone Lagrangian case by Biran-Cornea.
    c. Ramifications. This concerns a number of exciting relations with a number of different other subjects which are in the process of being understood today. For instance, relations with number theory as exemplified by recent work of McDuff-Schlenk as well as Biran-Cornea. Relations with toric geometry as described in the work of Fukaya-Oh-Ohta-Ono.


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