Abstract

All sorts of algebro-geometric moduli spaces (of stable curves, stable sheaves on a CY 3-foldsflat bundles, Higgs bundles..) are best understood as objects in derived geometry. Derivedenhancements 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 formore advanced constructions, such as categorification of enumerative invariants and (algebraic)deformation guantization of derived symplectic structures.l will outline how to construct suchenhancements for moduli spaces in global analysis and mathematical physics -that is. solutionspaces of nonlinear PDEs- in the framework of derived differential geometry and discuss the ellipticrepresentability theorem, which quarantees that, for elliptic equations, these derived moduli stacksare bona fide geometric objects (Artin stacks at worst). lf time pemmits l'll discuss applications toenumerative geometry (symplectic Gromov-Witten and Floer theory) and derived symplecticgeometry (the global BV formalism).