Abstract

This expository talk surveys several established ideas in derived algebraic geometry and their applications to enumerative geometry. I will begin with a concrete local introduction to derived schemes using commutative differential graded algebras, with derived intersections as guiding examples. This viewpoint naturally leads to perfect obstruction theories and virtual fundamental classes, two of the central tools in modern enumerative geometry. The derived language also makes it possible to formulate symplectic geometry in shifted cohomological degrees. The main focus will then be (-1)-shifted symplectic geometry: derived critical loci provide its basic local models, while moduli spaces of sheaves on Calabi--Yau threefolds serve as the central global examples. I will explain how (-1)-shifted symplectic structures, through the theory of vanishing cycles, refine numerical Donaldson--Thomas invariants into cohomological Donaldson--Thomas theory. Finally, I will briefly survey some recent developments concerning BPS cohomology and its connections with moduli spaces of Higgs bundles.

Organizer

Hsueh-Yung Lin (NTU)