Algebraic combinatorics, matrix integrals and algebraic geometry

组合学是研究有限对象集合的排列以及它们之间的关系的学科。组合学的研究主要受到其他数学和物理科学以及工程学应用的推动。许多这样的应用只需要确定集合中对象的数量，称为对集合进行计数。在代数组合学中，我们广泛使用了代数中的结果和方法。最基本的组合对象之一称为排列，它对有限集合中的元素进行重新排序。换位是一个非常简单的排列，它简单地交换(“换位”)两个元素的顺序。排列的乘积是通过逐次重新排序得到的，在代数中，一个集合的所有排列与这个乘积的集合称为对称群。本研究的目的是研究代数组合学在矩阵积分和代数几何中的应用，以及在最近的研究文献中特别重要的问题，因为它们的解决方案涉及到数学和物理中令人惊讶的各种领域。例如，在几何学中，赫维茨数作为球体的分支覆盖的个数出现，但在纯粹的组合术语中，这等同于给定的置换可以被表示为对称群中的换位与一定的连通性条件的乘积以及乘积中给定的置换的乘积的方法的数目。这些数字和一个新的变种在这个提议中被研究，并且最近已经引起了极大的研究兴趣，因为它们在二维重力、可积族、矩阵积分、模空间、计数几何和组合学的研究中具有重要意义。这一建议的主要兴趣在于使代数组合学和几何之间的相互作用成为双向的，而不是简单地解决所述的几何问题，而是更进一步地获得关于原始几何问题本身的附加信息，或者在几何范围内提出新的解决方法。

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.