Jang Soo Kim: Combinatorics of the Selberg integral.
In 1944, Selberg evaluated a multivariate integral, which generalizes Euler's beta integral. In 1980, Askey conjectured a $q$-integral version of the the Selberg integral, which was proved independently by Habsieger and Kadell in 1988. In this talk, we focus on the combinatorial aspects of the Selberg integral. First, we review the following fact observed by Igor Pak: evaluating the Selberg integral is essentially the same as counting the linear extensions of a certain poset. Considering $q$-integrals over order polytopes, we give a combinatorial interpretation for Askey's $q$-Selberg integral. We also find a connection between the Selberg integral and Young tableaux. As applications we enumerate Young tableaux of various shapes.