Differential Geometric PDE Moduli Spaces: Derived Enhancements, Ellipticity and Representability

All sorts of algebro-geometric moduli spaces (of stable curves, stable sheaves on a CY 3-folds, flat bundles, Higgs bundles...) are best understood as objects in derived geometry. Derived enhancements of classical moduli spaces give transparent and intrinsic meaning to previously ad-hoc structures pertaining to, for instance, enumerative geometry and are indispensable for more for more advanced constructions, such as categorification of enumerative invariants and (algebraic) deformation quantization of derived symplectic structures. I will outline how to construct such enhancements for moduli spaces in global analysis and mathematical physics -that is, solution spaces of nonlinear PDEs- in the framework of derived differential geometry and discuss the elliptic representability theorem, which guarantees that, for elliptic equations, these derived moduli stacks are bona fide geometric objects (Artin stacks at worst). If time permits, I'll discuss applications to enumerative geometry (symplectic Gromov-Witten and Floer theory) and derived symplectic geometry (the global BV formalism).