Algebraic combinatorics, matrix integrals and algebraic geometry

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Combinatorics is the study of arrangements of sets of finite objects and relationships between them. Research in combinatorics is centrally motivated by applications in other mathematical and physical sciences, and in engineering. Many such applications require only that one determine the number of objects in a set, called "counting" the set. In algebraic combinatorics we make extensive use of results and methods from algebra in this study. One of the most fundamental combinatorial objects is called a permutation, which reorders the elements of a finite set. A transposition is a very simple permutation, that simply interchanges the order of ("transposes") two of the elements. A product of permutations is obtained by successive reordering, and in algebra, the set of all permutations of a set with this product is called the symmetric group.The purpose of this Research Proposal is to study applications of algebraic combinatorics to matrix integrals and algebraic geometry, to problems that have been especially significant in the recent research literature because of the surprising variety of areas in mathematics and physics that their solutions involve. For example, Hurwitz numbers arise in geometry as the number of ramified covers of the sphere, but this is equivalent in purely combinatorial terms to the number of ways in which a given permutation can be expressed as a product of transpositions in the symmetric group with a certain "connectivity" condition, and a given number of permutations in the product. These numbers and a new variant are studied in this proposal, and have been of great recent research interest because they have been significant in the study of 2-dimensional gravity, integrable hierarchies, matrix integrals, moduli spaces, enumerative geometry, and combinatorics. Of major interest in this proposal is to make the interaction between algebraic combinatorics and geometry two-way, not simply solving a geometric problem as stated, but to go further and obtain additional information about the original geometric questions themselves, or suggest new methods of solution within geometry.