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.

我们在日常生活的几乎每个方面都会遇到对称性：看着镜子中自己的脸，看着雪花从天而降，开车穿过桥梁。 对称生物在自然界的进化中持续存在，对称主角在艺术中被认为特别美丽，对称组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性编码为称为矩阵的矩形数字数组来方便地研究这组所有对称性。 从自然界中的对称对象到相关矩阵集合的过程是表示论数学领域的标志。因此，表示论将自然界对称性的复杂研究简化为众所周知的数学领域（称为线性代数）中的问题。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出现的许多对称结构的理解。该项目还为本科生直接参与数学研究提供了机会，其重点目标是支持数学领域女性的发展和招募。该项目将讨论代数、几何和非阿基米德局部域上的还原代数群的表示论两个主题。 首先，将几何群论技术应用于相关的 Bruhat-Tits 构建，研究人员将提供一种全局方法来理解任何仿射 Weyl 群的共轭类。 其次，这种几何视角也将用于重新解释彼得森同构的 K 理论推广，从仿射格拉斯曼的等变同调到有限旗簇的等变量子上同调。 具体目标包括根据底层有限 Weyl 群对仿射共轭类的完整描述，以及仿射格拉斯曼的等变 K 同调中舒伯特类的新代表。 因此，该研究将刺激表示论、算术几何、枚举几何、代数组合学、几何群论和数学物理等数学子领域之间的新相互作用。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。