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Tuesday, November 19, 2013

**Abstract:** I will discuss families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their intrinsic connections to combinatorial q-series and automorphic forms. The generating functions for these families naturally lead to fundamental identities for theta functions and Hickerson's universal mock theta function. This provides a very general answer to a conjecture of Andrews, in which he predicted the modularity of the generating function for Schur's partitions. As a final application, we prove the striking result that the universal mock theta function can be expressed as a conditional probability in a certain natural probability space with an in finite sequence of independent events.