Combinatorial Group Actions and Applications to Geometry, Knot Theory, and Representation Theory

组合群动作及其在几何、纽结理论和表示论中的应用

• 批准号: 2054513
• 负责人: Julianna Tymoczko
• 金额: $ 28.66万
• 依托单位: Smith College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054513&HistoricalAwards=false
• 关键词: Combinatorial; Group; Actions; Applications; Geometry

Rotating a molecule without changing the chemical bonds it can form; braiding strands of hair, which is how quantum mechanics describes the structure of certain elementary particles; rearranging the nucleotides that make up the building blocks of a strand of DNA. All of these—rotations, braids, rearrangements—are examples of group actions, namely families of symmetries satisfying structural constraints that arise from the context of each application. This project studies group actions using combinatorial tools, including graphs and Young tableaux, in order to characterize their underlying algebraic structures. Applications include other areas of mathematics (like knot theory and algebraic geometry) and other fields (like mathematical biology). The broader impacts of the project include continued leadership of the Center for Women in Math at Smith College, mentoring of students at all levels through the PI's Math Research Lab as well as professional development for graduate students and continued career advice for alumni of the Center for Women in Math.More technically, this project addresses a set of interconnected questions about two graph-theoretic actions of permutations: 1) a skein-type action of permutations on a family of graphs called webs (of which noncrossing matchings are the best-known example), and 2) a representation constructed from graph automorphisms for a family of subgraphs of the Cayley graph of the permutation group. Each action is the nexus between several mathematical fields, giving rise to a cycle of related questions. In the first case, these are: combinatorially analyzing the transition matrices between two important bases of irreducible symmetric-group representations; describing the geometry and topology of components of Springer fibers; and constructing a basis for a vector space arising from questions in knot- and representation-theory. In the second case, they are: attacking the Stanley-Stembridge conjecture in combinatorics using a newly-established geometric correspondence; computing the (equivariant) cohomology rings of Hessenberg varieties; and describing generalized splines for different edge-labeled graphs. Recent work of Brosnan and Chow, Rhoades, and others mean that the field is poised for new advances. The project exploits a synergistic strategy by attacking these interrelated questions simultaneously, and uses computational aspects of the work to incorporate mentoring and practical training of students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

旋转分子而不改变它可以形成的化学键;编织头发，这是量子力学描述某些基本粒子结构的方式;重新排列构成DNA链的核苷酸。 所有这些-旋转，辫子，引用-都是群作用的例子，即满足每个应用环境中产生的结构约束的对称性族。 该项目使用组合工具研究群体行动，包括图形和Young tableaux，以描述其潜在的代数结构。 应用包括其他数学领域（如纽结理论和代数几何）和其他领域（如数学生物学）。该项目的更广泛的影响包括继续领导史密斯学院的女性数学中心，通过PI的数学研究实验室指导各级学生，以及研究生的专业发展和女性数学中心校友的持续职业建议。更技术性地说，该项目解决了一组关于排列的两个图论行为的相互关联的问题：1）排列在称为网的图族上的绞型作用（其中非交叉匹配是最著名的例子），以及2）由排列群的凯莱图的一族子图的图自同构构造的表示。每个动作都是几个数学领域之间的联系，产生了一个相关问题的循环。在第一种情况下，这些是：组合分析之间的过渡矩阵的两个重要基地的不可约的群表示;描述几何和拓扑结构的组件的施普林格纤维;和构建一个基础的向量空间所产生的问题，在结和代表性理论。在第二种情况下，他们是：攻击斯坦利-Stembridge猜想在组合数学中使用新建立的几何对应;计算（等变）上同调环的Hessenberg品种;和描述广义样条不同的边缘标记的图。Brosnan和Chow、Rhoades等人最近的工作意味着该领域有望取得新的进展。 该项目通过同时解决这些相互关联的问题，利用协同策略，并利用工作的计算方面，将指导和学生的实践培训结合起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。