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

在不改变分子可能形成的化学键的情况下旋转分子；编织头发，这是量子力学描述某些基本粒子的结构；重新排列组成DNA链的核苷酸。所有这些--旋转、辫子、重排--都是群体作用的例子，也就是对称族满足每个应用背景产生的结构约束。这个项目使用组合工具，包括图和Young Tableaux来研究群行动，以表征其潜在的代数结构。应用包括数学的其他领域(如纽结理论和代数几何)和其他领域(如数学生物学)。该项目的更广泛影响包括继续领导史密斯学院女性数学中心，通过Pi的数学研究实验室指导所有级别的学生，以及研究生的专业发展和为数学女性中心的校友提供持续的职业建议。更严格地说，该项目解决了一组相互关联的问题，这些问题涉及两个关于排列的图论行为：1)称为网络的图族上排列的偏斜型行为(其中不交叉匹配是最著名的例子)，以及2)由图自同构构造的表示，用于置换群的Cayley图子图族。每个动作都是几个数学领域之间的纽带，引发了一个相关问题的循环。在第一种情况下，它们是：组合分析不可约对称群表示的两个重要基之间的转移矩阵；描述Springer纤维分量的几何和拓扑；以及构造由纽结和表示论中的问题引起的向量空间的基。在第二种情况下，它们是：利用新建立的几何对应来攻击组合学中的Stanley-Stembridge猜想；计算Hessenberg簇的(等变)上同调环；以及描述不同边标记图的广义样条线。布鲁斯南和周、罗兹等人最近的工作意味着，该领域即将取得新的进展。该项目通过同时解决这些相互关联的问题来利用协同策略，并使用工作的计算方面来结合对学生的指导和实践培训。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.