Random Partitions, Random Matrices, and Combinatorics of Moduli Spaces of Curves

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The proposer is going to study a number of interconnected combinatorial problems related, on the one hand, to exact or asymptotic enumeration of branched coverings of one surface by another and, on the other hand, to exact or asymptotic evaluation of various sums over partitions. These problems arise both from enumerative geometry of the moduli spaces of curves (and related moduli spaces) and from connection with random matrices. If solved, these problems will lead to important applications to algebraic geometry, ergodic theory, and other fields, and will also substantially further our present understanding of the interaction between geometry of the moduli spaces and combinatorics of random matrices and partitions.