Collaborative Research: FRG: Eigenvalue and Saturation Problems for Reductive Groups

合作研究：FRG：还原群的特征值和饱和问题

• 批准号: 0554247
• 负责人: Shrawan Kumar
• 金额: $ 29.65万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554247&HistoricalAwards=false
• 关键词: Collaborative; Research; FRG; Eigenvalue; Saturation

DMS 0554254, PI: John Millson, Co-PI: Thomas HainesDMS 0554349, PI: Michael KapovichDMS 0554247, PI: Shrawan Kumar, Co-PI: Prakash Belkale Research in Lie theory has undergone striking advances in a number of directions spurred by and giving rise to discoveries in topology, symplectic and algebraic geometry and combinatorics. The recent solutions by Klyachko of the eigenvalues of a sum of Hermitian matrices problem and by Knutson and Tao of the saturation and Horn conjectures are of particular relevance to this proposal. Both these problems are associated to the group of invertible n by n matrices. The discovery of quantum cohomology and the quantum Schubert calculus led to the solution of analogous problems for the group of unitary n by n matrices. The PIs P.Belkale, T. Haines, M. Kapovich, S. Kumar and J. Millson propose to attack for a general reductive group G the problems previously solved for the groups of invertible unitary n by n matrices. The history of Lie theory has shown that it is of critical importance to understand in the context of general Lie groups results proved initially for the group of invertible n by n matrices.A large part of mathematics and physics has been involved with the study of eigenvalues, for example determining the modes of vibration of a violin string or the energy levels of an atom amounts to finding eigenvalues of a Hermitian linear operator. A fundamental problem is to determine the possibilities for the eigenvalues of the sum of two operators given the eigenvalues of each one. Another fundamental problem with its roots in physics is the problem of studying the representations of an abstract group as a group of matrices. This problem in turn has been organized into subproblems. One of the most important of these is the problem of decomposing (tensor) products of representations as sums of representations. The point of this proposal is that these two basic problems, the eigenvalue of the sum problem and the decomposing (tensor) products problem are very closely related. The authors propose to pin down this relationship (already well-understood for special cases) for the general case. This FRG grant will play a fundamental role in the further development of a national group of scientists working on the area common to Lie theory, topology, algebraic and symplectic geometry, combinatorics and the theory of buildings. A class of young mathematicians especially graduate students and postdocs in the mid-Atlantic area (including Washington and Chapel Hill) and the greater (San Francisco) bay area (including Davis) will have the opportunity to learn about and work on exciting and fundamental problems through the Meetings and Workshops envisaged by the PIs. This class already includes the nine graduate students presently advised by the PIs. We expect to include other graduate students and postdocs associated to the very strong programs in representation theory and geometry at the University of Maryland, the University of North Carolina at Chapel Hill and the University of California at Davis. We also expect that graduate students, postdocs and faculty from neighboring universities such as Johns Hopkins, Duke, North Carolina State University, UC-Berkeley and Stanford will participate in and profit from the FRG grant. This award is jointly funded by the programs in Analysis, and Algebra, Number Theory, & Combinatorics, and Geometric Analysis.

DMS 0554254, PI: John Millson, Co-PI: Thomas HainesDMS 0554349, PI: Michael KapovichDMS 0554247, PI: Shrawan Kumar, Co-PI: Prakash Belkale 谎言理论的研究在许多方向上取得了惊人的进展引起拓扑学、辛几何、代数几何和组合学的发现。 Klyachko 最近对 Hermitian 矩阵问题和的特征值的解决方案以及 Knutson 和 Tao 对饱和度和霍恩猜想的解决方案与该提案特别相关。这两个问题都与可逆 n × n 矩阵群相关。量子上同调和量子舒伯特微积分的发现导致了酉 n × n 矩阵群的类似问题的解决。 PIs P.Belkale、T. Haines、M. Kapovich、S. Kumar 和 J. Millson 提议针对先前针对可逆酉 n × n 矩阵群解决的问题，提出攻击一般还原群 G 的问题。李理论的历史表明，在一般李群的背景下理解最初证明的可逆 n × n 矩阵群的结果至关重要。数学和物理学的很大一部分都涉及到特征值的研究例如，确定小提琴弦的振动模式或原子的能级相当于找到埃尔米特线性算子的特征值。一个基本问题是在给定每个运算符的特征值的情况下确定两个运算符之和的特征值的可能性。另一个源于物理学的基本问题是研究抽象群作为一组矩阵的表示的问题。这个问题又被组织成子问题。其中最重要的问题之一是将表示的（张量）乘积分解为表示之和的问题。该提案的要点是，求和问题的特征值和分解（张量）乘积问题这两个基本问题非常密切相关。作者建议为一般情况确定这种关系（对于特殊情况已经很好理解）。 FRG 的这笔资助将在国家科学家小组的进一步发展中发挥基础性作用，该小组致力于李理论、拓扑、代数和辛几何、组合数学和建筑理论的共同领域。大西洋中部地区（包括华盛顿和教堂山）和大（旧金山）湾区（包括戴维斯）的一类年轻数学家，特别是研究生和博士后，将有机会学习和研究令人兴奋的基本问题通过 PI 设想的会议和研讨会。该班已经包括目前由 PI 指导的九名研究生。我们希望包括与马里兰大学、北卡罗来纳大学教堂山分校和加州大学戴维斯分校的表示论和几何方面非常强大的项目相关的其他研究生和博士后。我们还期望约翰·霍普金斯大学、杜克大学、北卡罗来纳州立大学、加州大学伯克利分校和斯坦福大学等邻近大学的研究生、博士后和教师将参与 FRG 资助并从中受益。该奖项由分析、代数、数论、组合学和几何分析项目共同资助。