Algebraic combinatorics, matrix integrals and algebraic geometry

代数组合、矩阵积分和代数几何

• 批准号: 8907-2013
• 负责人: Goulden, Ian
• 金额: $ 2.48万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635562
• 关键词: Algebraic; combinatorics; matrix; integrals; algebraic

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.

组合学是对有限对象及其之间关系的布置的研究。组合学的研究是由其他数学和物理科学以及工程学中的应用中的集中动机。许多此类应用程序仅要求一个确定一个集合中的对象数量，称为“计数”集合。在代数组合学中，我们在本研究中广泛使用代数的结果和方法。最基本的组合对象之一称为排列，它可以重新定位有限集的元素。换位是一个非常简单的置换，它只是互换了（“转移”）两个元素的顺序。置换的产物是通过连续的重新排序获得的，在代数中，该产品的所有置换量都称为对称组。该研究建议的目的是研究代数组合对矩阵积分和代数几何学的应用，在最近的研究中尤其重要，因为该领域的实体是尤其重要的，因为该领域尤其重要，因为该领域的实体构成了众所周知的实现。例如，在几何形状中出现了Hurwitz数字作为球体的覆盖范围的数量，但这与纯粹组合术语相当于给定排列可以作为对称组中的转换产物表达为具有一定的“连接性”条件的对称组中的乘积，并且产品中给定的置换次数。在该提案中研究了这些数字和新的变体，并且具有极大的研究兴趣，因为它们在研究二维引力，可集成的层次结构，基质积分，模量空间，枚举几何形状和组合术方面具有重要意义。该提案的主要兴趣是使代数组合学和几何形状之间的相互作用双向进行，而不仅仅是解决几何问题，而是要进一步解决并获取有关原始几何问题本身的其他信息，或者建议在几何学中提出新的解决方案方法。