Organized as part of the IHÉS Lectures, this Summer School aims to provide PhD students, post-docs, and young researchers with an overview of recent developments in the theory of moduli spaces of pseudoholomorphic curves in symplectic and contact geometry. Moduli spaces of pseudoholomorphic curves arise as the zero set of a Fredholm section of a suitable bundle. Provided this section can be appropriately perturbed, these moduli spaces yield powerful contact and symplectic invariants such as Gromov-Witten theory, contact homology, symplectic homology, and Symplectic Field Theory. Constructions and applications of these invariants will be addressed in detail during the workshop. There are two main perturbative techniques, geometric and functional analytic. The geometric perturbation methods are powerful for applications and practical from a computational point of view but typically require many restrictive assumptions and fail to generalize broadly. We will introduce researchers to the polyfold machinery of Hofer, Wysocki, and Zehnder, a new analytic framework designed to resolve the issue of transversality systematically. As computations are integral in applications of the aforementioned invariants, we will also explore how geometric perturbation schemes can be incorporated into the polyfold package. We will supplement 7 mini courses with moderated discussions and related talks by senior faculty on current and future directions for the field.

Scientific Advisory Committee: H. Hofer (Institute for Advanced Study)
M. Hutchings (UC Berkeley)
D. McDuff (Barnard College, Columbia University)