**Monday, September 16, 2013** (4-5pm, Yost 306)

**Title: Elliptic hypergeometric integrals**
**Speaker:** Eric Rains (California Institute of Technology)

**Abstract:** Euler's beta (and gamma) integral lies at the core of much of the theory of special functions, and many generalizations have been studied, including multivariate analogues (the Selberg integral; also work of Dixon and Varchenko), $q$-analogues (Askey--Wilson, Nasrallah--Rahman), and both (work of Milne--Lilly and Gustafson). In 2001, van Diejen and Spiridonov conjectured several generalizations going beyond $q$ to the \emph{elliptic} level (replacing $q$ by a point on an elliptic curve). I'll discuss how a simple question in random matrix theory led to a proof of their conjectured identities, in fact generalizing them to a transformation (\emph{\`a~la} the integral representation of hypergeometric functions). I'll also discuss an elliptic Selberg integral with a (partial) symmetry under the Weyl group $E_8$, as well as connections with the theory of Macdonald and Koornwinder polynomials.