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

May-October 2015

THEMES

Many areas of modern geometry lead naturally to moduli spaces classifying certain geometric objects and a minimality problem on these moduli spaces. For instance, Perelman's proof of the Poincaré conjecture uses the Ricci flow, hence a parabolic evolution equation on the moduli space of metrics. The same setting is also present in higher dimensions, where Aubin, Yau, Tian and Donaldson established fundamental existence theorems. In Contact-Symplectic topology, the focus of the program, the same concepts are also behind the recent spectacular proof by Taubes of the full Weinstein conjecture on the existence of closed orbits of the Reeb flow on contact manifolds, which uses implicitly Seiberg-Witten theory hence moduli spaces of monopoles and the Embedded Contact Homology developed by Hutchings. The ubiquitous Floer theory is present almost everywhere in symplectic topology, but has also found extraordinary applications in low-dimensional differential topology. The theory of $J$-holomorphic curves, which is the core of the Gromov-Witten theory, has been used in almost complex geometry at all levels, but also by Welschinger, Kharlamov, Itenberg and Salomon to derive new real enumerative invariants. The proposed theme semester at CIRM will be a hub dedicated to the study of these questions that lie at the heart of the current developments in symplectic and differential topology, as plenary talks at the ICM's 1998, 2002, 2006 and 2010 show.