RUI: Geometry of Conjugacy and K-Theory in Affine Weyl Groups

RUI：仿射外尔群中的共轭几何和 K 理论

• 批准号: 2202017
• 负责人: Elizabeth Milicevic
• 金额: $ 17万
• 依托单位: Haverford College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202017&HistoricalAwards=false
• 关键词: RUI; Geometry; Conjugacy; Theory; Affine

We encounter symmetry in nearly every aspect of our daily lives: looking at our faces in the mirror, watching snowflakes fall from the sky, and driving across bridges. Symmetric organisms persist through evolution in nature, symmetric protagonists are perceived as especially beautiful in art, and symmetric components are critical to engineering structures that can withstand powerful forces. The set of symmetries of a particular physical object enjoys a rich algebraic structure, because symmetries are operations that can be composed together. This group of all symmetries can then be conveniently studied by encoding each symmetry as a rectangular array of numbers called a matrix. This process of passing from a symmetric object in the natural world to a related collection of matrices is the hallmark of the mathematical field of representation theory. Representation theory thus reduces the complex study of symmetry in nature to questions in the well understood area of mathematics called linear algebra. As such, the proposed projects have broad potential to substantially impact our understanding of many symmetric structures occurring throughout the mathematical and natural sciences. This project also provides opportunities for directly involving undergraduate students in mathematical research, with a focused goal of supporting the development and recruitment of women in mathematics.This project will address two topics in the algebra, geometry, and representation theory of reductive algebraic groups over non-archimedean local fields. First, applying techniques from geometric group theory to the associated Bruhat-Tits building, the investigator will provide a global approach to understanding the conjugacy classes of any affine Weyl group. Second, this geometric perspective will also be applied to reinterpret the K-theoretic generalization of Peterson's isomorphism from the equivariant homology of the affine Grassmannian to the equivariant quantum cohomology of a finite flag variety. Specific objectives include a complete description of an affine conjugacy class in terms of the underlying finite Weyl group, and new representatives for the Schubert classes in the equivariant K-homology of the affine Grassmannian. As such, the research will stimulate new interactions among the mathematical subfields of representation theory, arithmetic geometry, enumerative geometry, algebraic combinatorics, geometric group theory, and mathematical physics.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.

我们几乎在日常生活的每一个方面都遇到了对称：看着镜子中的自己的脸，看着雪花从天而降，开车过桥。对称的生物体在自然界的进化中一直存在，对称的主角在艺术中被认为是特别美丽的，对称的部件对于能够承受强大力量的工程结构至关重要。特定物理对象的对称性集合具有丰富的代数结构，因为对称性是可以组合在一起的运算。然后，可以通过将每个对称编码为称为矩阵的矩形数字数组来方便地研究所有对称性的这组。从自然界中的一个对称物体到一个相关的矩阵集合的这个过程是表示论数学领域的标志。因此，表示理论将对自然界中对称性的复杂研究归结为被称为线性代数的数学领域中众所周知的问题。因此，拟议的项目具有广泛的潜力，可以极大地影响我们对数学和自然科学中许多对称结构的理解。这个项目还提供了直接让本科生参与数学研究的机会，重点是支持数学领域女性的发展和招聘。这个项目将讨论非阿基米德局部域上的代数、几何和约化代数群的表示理论的两个主题。首先，将几何群论的技巧应用到相关的Bruhat-Tits建筑中，调查者将提供一种全局方法来理解任何仿射Weyl群的共轭类。其次，这个几何观点也将被用来重新解释K-理论对Peterson同构的推广，从仿射Grassmanian的等变同调到有限FLAG簇的等变量子上同调。具体目标包括用基础有限Weyl群的形式完整地刻画仿射共轭类，以及在仿射Grassmanian的等变K-同调中Schubert类的新的表示。因此，这项研究将促进表示论、算术几何、计数几何、代数组合学、几何群论和数学物理等数学子领域之间的新互动。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。