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CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, October 23, 2014 - 2:30pm

Jonathan Novak

MIT

Location

University of Pennsylvania

DRL 4E9

This talk is about uniformly random lozenge tilings of a special class of planar domains, which I like to call "sawtooth domains." Sawtooth domains are special in that their tilings are in bijection with Gelfand-Tsetlin patterns (or semistandard Young tableaux), and consequently many of their observables can be expressed in terms of special functions associated to the representation theory of GL(N), e.g. Schur functions. I will describe a new approach to random tilings in which Schur functions are replaced by an equivalent object, the Harish-Chandra/Itzykson-Zuber matrix integral (the equivalence between Schur and HCIZ is an instance of the Kirillov character formula). Remarkably, the HCIZ integral is a generating function for a certain desymmetrization of the Hurwitz numbers from classical enumerative geometry, and this fact allows one to read off limit theorems for random tilings, bypassing the usual contour integrals and steepest descent analysis.