### Moduli spaces and elliptic difference equations

Rains, EM *(CALTECH)*

Wednesday 01 July 2009, 09:00-10:00

Seminar Room 1, Newton Institute

#### Abstract

Sakai's elliptic Painlevé equation is constructed via the geometry of certain rational algebraic surfaces. I'll discuss joint work with Arinkin and Borodin in which we interpret Sakai's surfaces as moduli spaces of second-order linear elliptic difference equations (and related objects), such that the elliptic Painlevé equation itself acts via isomonodromy transformations. The construction also lends itself to higher-order problems, and/or problems with more complicated singularity structure; I'll discuss what is known about the corresponding higher-dimensional moduli spaces.

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