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

我们在日常生活的方方面面几乎都会遇到对称：照镜子看自己的脸，看天上飘落的雪花，开车过桥。对称的生物体在自然界的进化中持续存在，对称的主角在艺术中被认为是特别美丽的，对称的组件对于能够承受强大力量的工程结构至关重要。一个特定物理对象的对称集合具有丰富的代数结构，因为对称是可以组合在一起的操作。通过将每个对称编码为称为矩阵的矩形数字数组，可以方便地研究这组所有对称。这种从自然界的对称对象传递到相关矩阵集合的过程是表征理论数学领域的标志。表征理论因此将自然界对称性的复杂研究简化为被称为线性代数的数学领域的问题。因此，提议的项目具有广泛的潜力，可以实质性地影响我们对整个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，其重点目标是支持女性在数学领域的发展和招聘。本计画将讨论代数、几何和非阿基米德局部域上约化代数群的表示理论两个主题。首先，应用从几何群论到相关Bruhat-Tits构建的技术，研究者将提供一个理解任何仿射Weyl群的共轭类的全局方法。其次，这一几何视角也将被应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标志变体的等变量子上同调。具体的目标包括在潜在的有限Weyl群中对仿射共轭类的完整描述，以及仿射Grassmannian的等变k -同调中的Schubert类的新代表。因此，这项研究将激发表征理论、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互动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.