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

提议者将研究一些相关的相互关联的组合问题，一方面，精确或渐近枚举的分支覆盖的一个表面由另一个，另一方面，精确或渐近评估的各种总和分区。 这些问题都产生于曲线的模空间（和相关的模空间）的枚举几何和随机矩阵的连接。 如果得到解决，这些问题将导致重要的应用代数几何，遍历理论，和其他领域，也将大大促进我们目前的理解之间的相互作用的几何模空间和组合的随机矩阵和分区。

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.