Mathematics Colloquia and Seminars

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Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius, and diameter) of a "typical" compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; by a theorem of Belyi these are "dense" in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichm\"{u}ller spaces depends on the study of oriented cycles in random regular graphs with random orientation. Brooks and Makover conjectured that asymptotically normalized cycles lengths follow Poisson-Dirichlet distribution. We will present a proof of this conjecture using representation theory of the symmetric group.