I will explain a simple reformulation of the Hochschild–Kostant–Rosenberg theorem, describing the algebra of (crystalline) differential operators on a smooth scheme over a commutative ring k in terms of (higher) Hochschild cohomology. When k is replaced by a commutative ring spectrum, essentially the same construction gives a way of assigning to each 1-dimensional formal group H an algebra of “H-differential operators” on affine space. When H is the additive (resp. multiplicative) formal group, this recovers usual differential operators (resp. q-difference operators). Computing the algebra of global H-difference operators on P^1 leads to a definition of an associative algebra U_H(PGL_2) which recovers the usual enveloping algebra (resp. quantum group) when H is the additive (resp. multiplicative) formal group.
Many aspects of ordinary (and q-) calculus generalize to H-differential operators (and admit nice homotopy-theoretic explanations), like the Gauss–Manin connection, Frobenius-constant quantizations via p-curvature, Lagrangian p-support, Mellin + Fourier transforms, “H-analogues” of various special functions like hypergeometric functions, etc. Since these objects are defined using homotopy theory, one can in fact give topological proofs of various identities involving these special functions! This story has two (mirror dual!) motivations: relative Langlands duality + quantum generalized cohomology; and the theory of prismatic cohomology. If there is time, I will discuss applications to both. Some of this story builds on joint work with various subsets of Jeremy Hahn, Max Misterka, Arpon Raksit, and Allen Yuan.