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

我们几乎在日常生活的每一个方面都遇到了对称：在镜子中看着我们的脸，看着雪花从天空中飘落，开车过桥。 对称的有机体在自然界的进化过程中一直存在，对称的主角在艺术中被认为特别美丽，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。 所有对称的这个群可以通过将每个对称编码为称为矩阵的矩形数组来方便地研究。 这个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，表示论将对自然界中对称性的复杂研究简化为被称为线性代数的数学领域中的问题。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，其重点目标是支持数学领域的妇女发展和招聘，该项目将涉及代数、几何和非阿基米德局部域上的约化代数群的表示论两个主题。 首先，应用技术从几何群论相关的Bruhat-Tits建设，调查员将提供一个全球性的方法来理解任何仿射Weyl群的共轭类。 其次，这种几何的角度也将被应用到重新解释的K-理论推广彼得森的同构从等变的同源性的仿射格拉斯曼的等变量子上同调的有限旗品种。 具体目标包括一个完整的描述的仿射共轭类的基本有限Weyl组，和新的代表舒伯特类的等变K-同源的仿射格拉斯曼。 因此，该研究将激发表示论，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的相互作用。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.