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

DMS 0554254, PI: John Millson, Co-PI: Thomas HainesDMS 0554349, PI: Michael KapovichDMS 0554247, PI: Shrawan Kumar， Co-PI: Prakash Belkale在拓扑学、辛代数几何和组合学的发现的推动下，李论的研究在许多方向上取得了惊人的进展。最近由Klyachko提出的厄米矩阵和问题的特征值的解以及Knutson和Tao提出的饱和猜想和Horn猜想的解都与这一建议特别相关。这两个问题都与可逆n × 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n × n酉矩阵群的类似问题的解决。P.Belkale， T. Haines， M. Kapovich， S. Kumar和J. Millson提出了对一般约化群G的攻击，解决了以前可逆的n × n酉矩阵群所解决的问题。李论的历史表明，在一般李群的背景下理解最初为可逆n × n矩阵群证明的结果是至关重要的。数学和物理学的很大一部分已经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厄米线性算子的特征值。一个基本问题是确定两个算子的和的特征值的可能性，给定每个算子的特征值。另一个植根于物理学的基本问题是研究抽象群作为矩阵群的表示问题。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）乘积分解为表示的和。这一建议的要点在于，和的特征值问题和分解（张量）积问题这两个基本问题是密切相关的。作者建议在一般情况下确定这种关系（在特殊情况下已经得到了很好的理解）。这项FRG拨款将在一个国家科学家小组的进一步发展中发挥重要作用，该小组致力于李理论、拓扑、代数和辛几何、组合学和建筑理论等领域的共同研究。一群年轻的数学家，尤其是大西洋中部地区（包括华盛顿和教堂山）和大湾区（包括旧金山）的研究生和博士后，将有机会通过pi设想的会议和研讨会来学习和研究令人兴奋的基本问题。这个班已经包括了目前由pi指导的九名研究生。我们希望包括马里兰大学、北卡罗来纳大学教堂山分校和加州大学戴维斯分校的其他研究生和博士后，他们都与非常强大的表示理论和几何项目有关。我们也期待来自邻近大学的研究生、博士后和教职员工，如约翰霍普金斯大学、杜克大学、北卡罗莱纳州立大学、加州大学伯克利分校和斯坦福大学，将参与并从FRG资助中获益。该奖项由分析、代数、数论、组合学和几何分析项目共同资助。

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.