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

DMS 0554254，PI：John Millson，Co-PI：Thomas Haines DMS 0554349，PI：Michael Kapovich DMS 0554247，PI：Shrawan Kumar，Co-PI：Prakash Belkale李理论研究在拓扑学、辛几何、代数几何和组合学发现的刺激下，在许多方向上取得了惊人的进展。最近Klyachko对厄米特矩阵之和问题的特征值的解，以及Knutson和Tao对饱和猜想和Horn猜想的解与这一建议特别相关。这两个问题都与n乘n矩阵的可逆组有关。量子上同调和量子Schubert演算的发现导致了酉n×n矩阵群的类似问题的解决。P.Belkale，T.Haines，M.Kapovich，S.Kumar和J.Millson等人建议对一般可约群G攻击以前解决的n×n矩阵可逆群的问题。李论的历史表明，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。很大一部分数学和物理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等于寻找厄米特线性算子的特征值。一个基本的问题是确定给定每个算子的特征值的两个算子之和的特征值的可能性。另一个根植于物理学的基本问题是研究抽象群的表示为一组矩阵的问题。这个问题又被组织成了子问题。其中最重要的问题之一是将表示的(张量)积分解为表示和的问题。这一建议的要点是，和问题的特征值和分解(张量)积问题这两个基本问题是密切相关的。作者建议将这种关系(在特殊情况下已经被很好地理解)限定在一般情况下。这笔FRG赠款将在一个致力于李理论、拓扑学、代数和辛几何、组合学和建筑理论共同领域的国家科学家小组的进一步发展中发挥重要作用。一班年轻的数学家，特别是大西洋中部地区(包括华盛顿和教堂山)和大湾区(包括戴维斯)的研究生和博士后，将有机会通过私人投资促进机构设想的会议和讲习班，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私人投资顾问指导的9名研究生。我们预计将包括马里兰大学、北卡罗来纳大学教堂山分校和加州大学戴维斯分校非常强大的表示理论和几何专业的其他研究生和博士后。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并受益于FRG赠款。该奖项由分析、代数、数论、组合数学和几何分析等课程共同资助。

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.