Algebraic combinatorics, matrix integrals and algebraic geometry

组合数学是研究有限对象集合的排列以及它们之间的关系。组合学的研究主要是由其他数学和物理科学以及工程学的应用所推动的。许多这样的应用程序只需要确定一个集合中对象的数量，称为“计数”集合。在代数组合学中，我们广泛使用代数的结果和方法进行研究。最基本的组合对象之一被称为置换，它对有限集合的元素进行重新排序。转置是一种非常简单的排列，它只是简单地交换（“转置”）两个元素的顺序。置换的乘积是通过连续的重新排序得到的，在代数中，具有该乘积的集合的所有置换的集合被称为对称群。 本研究计划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，这些问题在最近的研究文献中特别重要，因为它们的解决方案涉及数学和物理领域的惊人变化。例如，赫维茨数在几何学中作为球面的分支覆盖数出现，但这在纯粹的组合术语中等价于一个给定的置换可以表示为对称群中具有某种“连通性”条件的置换与乘积中给定数量的置换的乘积的方式的数量。这些数字和一个新的变种研究在这个建议中，并已被极大的最近的研究兴趣，因为他们一直在研究二维重力，可积层次，矩阵积分，模空间，枚举几何和组合学的重要。在这个建议中主要的兴趣是使代数组合学和几何之间的相互作用是双向的，而不是简单地解决所述的几何问题，而是进一步获得关于原始几何问题本身的额外信息，或者建议几何中的新的解决方法。

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.