Deformation Quantization (Calculus) and Vanishing Cycles (Topology)

Functions with no antiderivative detect the presence of holes in a domain; this is a basic manifestation of a deep relationship between calculus and topology. This talk will survey some classical results in this vein (de Rham's theorem, the Riemann-Hilbert correspondence) moving towards a slightly more sophisticated manifestation which features in ongoing joint work with Pavel Safronov: a correspondence between deformation quantization and a certain perverse sheaf of local vanishing cycles.