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: 0554349
• 负责人: Michael Kapovich
• 金额: $ 25.87万
• 依托单位: University of California-Davis
• 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 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554349&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：托马斯海恩斯DMS 0554349，PI：Michael KapovichDMS 0554247，PI：Shrawan Kumar，Co-PI：Prakash Belkale李理论的研究在拓扑学、辛几何学、代数几何学和组合学的发现的推动下，在许多方向上都取得了惊人的进展。最近的解决方案Klyachko的特征值的总和厄米特矩阵的问题和克努森和陶的饱和度和霍恩矩阵是特别相关的这一建议。这两个问题都与可逆n乘n矩阵群有关。量子上同调和量子舒伯特演算的发现导致了酉n乘n矩阵群的类似问题的解决。P.Belkale，T.海恩斯，M. Kapovich，S. Kumar和J. Millson提出了对一般约化群G的攻击，以前解决了可逆酉n乘n矩阵群的问题。李群理论的历史表明，在一般李群的背景下理解最初证明的可逆n乘n矩阵群的结果是至关重要的。数学和物理学的很大一部分都涉及本征值的研究，例如确定小提琴弦的振动模式或原子的能级相当于找到厄米特线性算子的本征值。一个基本的问题是确定的可能性的总和两个运营商的本征值的每一个。另一个根源于物理学的基本问题是研究抽象群作为矩阵群的表示的问题。这个问题又被组织成子问题。其中最重要的一个问题是将表示的（张量）乘积分解为表示的和的问题。这个建议的要点是，这两个基本问题，和问题的特征值和分解（张量）产品问题是非常密切相关的。作者建议在一般情况下确定这种关系（在特殊情况下已经很好理解）。这联邦德国赠款将发挥根本性的作用，在进一步发展的一个国家组的科学家工作的共同领域李理论，拓扑，代数和辛几何，组合数学和理论的建筑物。大西洋中部地区（包括华盛顿和查佩尔山）和大（弗朗西斯科）湾区（包括戴维斯）的一类年轻数学家，尤其是研究生和博士后，将有机会通过PI设想的会议和研讨会了解和研究令人兴奋的基本问题。这门课已经包括了目前由PI建议的九名研究生。我们希望包括其他研究生和博士后相关的非常强大的程序在表示理论和几何在马里兰州，北卡罗来纳州大学在查佩尔山和加州大学戴维斯分校。我们还预计来自约翰·霍普金斯大学、杜克大学、北卡罗来纳州州立大学、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联邦德国政府的资助并从中受益。该奖项由分析，代数，数论，组合数学和几何分析课程共同资助。