RUI: Affine Flags, p-adic Representations, and Quantum Cohomology

该研究项目解决代数几何中的问题，研究多项式方程组的解决方案，以及表示论，旨在解释数学和自然科学中对称性的基本构建块。本计画的主要研究对象是具有幂级数元素的可逆矩阵群。这种局部域上的代数群有一个特别漂亮的分解，分解成由一组变换的元素索引的单元，这些变换是由欧几里得空间中超平面的反射产生的。这种细胞分解允许的方法来理解的代数几何和表示理论的矩阵组采用组合和几何技术，利用丰富的对称性的安排功能的反射超平面。该项目还提供涉及本科生的数学研究，通过夏季研究计划，长达一年的论文项目，并参加当地的座谈会，区域研讨会和全国性conferences.The调查人员将利用和扩展p-adic表示理论，仿射旗品种之间的令人惊讶的关系在积极的特点，量子上同调复杂的格拉斯曼，和仿射格拉斯曼的同源性。具体的研究目标包括明确的无类型公式的尺寸仿射Deligne-Lusztig品种，值的p-adic轨道积分，量子和仿射舒伯特类的产品。大多数项目的主要工具是仿射旗簇的壁龛行走模型，这是一个统一的组合平台，将仿射Hecke代数，晶体基，Mirkovic-Vilonen循环，量子和仿射Schubert微积分以及Kac-Moody群构建中的测地线的研究联系起来。

This research project addresses problems in algebraic geometry, which studies solutions to systems of polynomial equations, and in representation theory, which aims to explain the basic building blocks of symmetry in mathematics and natural science. The central objects of study in this project are groups of invertible matrices with power series entries. Such algebraic groups over local fields have an especially beautiful decomposition into cells indexed by elements of a group of transformations that is generated by reflections across hyperplanes in Euclidean space. This cell decomposition permits an approach to understanding the algebraic geometry and representation theory of the matrix group by employing combinatorial and geometric techniques that exploit the abundant symmetry featured in the arrangement of the reflecting hyperplanes. The project also provides involves undergraduate students in mathematical research through summer research programs, year-long thesis projects, and participation in local colloquia, regional seminars, and national conferences.The investigator will utilize and extend surprising relationships among p-adic representation theory, affine flag varieties in positive characteristic, the quantum cohomology of complex Grassmannians, and the homology of the affine Grassmannian. Concrete goals of the research include explicit type-free formulas for dimensions of affine Deligne-Lusztig varieties, values of p-adic orbital integrals, and products of quantum and affine Schubert classes. The primary tool in most projects is the alcove walk model for the affine flag variety, a uniform combinatorial platform that connects the study of affine Hecke algebras, crystal bases, Mirkovic-Vilonen cycles, quantum and affine Schubert calculus, and geodesics in the building of Kac-Moody groups.