Moduli spaces of differential and difference equations -- Eric Rains, April 1, 2011

One of the striking properties of the differential equations satisfied by hypergeometric functions is rigidity: up to isomorphism, they are uniquely determined by their behaviour at the singular points. Similarly, the "space of initial conditions" of a Painlevé equation can be identified as a moduli space of differential equations with fixed singularity structure. This motivates us to study such moduli spaces in general, including difference (ordinary, q-, or elliptic) equations. We show that such spaces are symplectic, are rational under relatively mild assumptions, and determine precisely which singularity structures are rigid. The key tool is an identification with an open subset of a moduli space of 1-dimensional sheaves on an appropriate rational surface, allowing one to apply geometric ideas. For instance, rigid singularity structures map to -2 curves, and the induced action of birational transformations on such sheaves gives rise to generalizations of the Fourier transform. (Parts are joint work with Arinkin and Borodin, other parts joint with Okounkov)