Derived Symplectic Geometry and Applications (DerSympApp)
A project funded by the European Research Council (Consolidator Grant 768679)
Duration: Sept. 2018 - Aug. 2023
Symplectic geometry is a natural setting for classical mechanics: phase spaces appear to be symplectic manifolds (or variations of these). Indeed, a symplectic (or, rather, Poisson) structure is precisely what is needed to associate a dynamical system on the given space to each energy function.
Derived algebraic geometry has been invented in order to deal in a satisfying way with spaces that are singular, which is the case for many spaces appearing in algebraic geometry (moduli spaces) and in classical physics (spaces of solutions of equations of motion).
The theory of symplectic structures on derived stacks gave birth to what is now called derived symplectic geometry. Since then we have seen a rapid evolution of knowledge, and several applications as well (such as general existence theorems for symplectic structures and symmetric obstruction theories on moduli spaces, for instance). This project aims at providing new theoretical developments (shifted symplectic groupoids, for example) and new applications of derived symplectic geometry (to classical field theories, for instance).