Horn's Conjecture and related Problems

摘要自20世纪90年代中期Klyashko的重大突破以来，在表示理论、代数几何和组合学中与Horn猜想相关的广泛问题引起了极大的兴奋和新的兴趣。特别是，Balkale, Buch, Fulton, Knutson, Tao， Woodward和许多其他人都获得了重要的结果。相比之下，Horn猜想在算子理论中并没有引起那么多的关注，即使这个猜想是用自伴随矩阵来表述的。霍恩猜想的证明使用了来自代数几何和组合学的高度非平凡的工具。在过去的十年里，Bercovici和PI一直在进行一个项目，寻找一个可以推广到冯·诺伊曼代数设置的霍恩猜想的建设性证明。最近的进展显示出新的希望。本计画将提供对特征值问题和特征旗复杂几何形状的新认识。PI和她的合作者还计划使用他们迄今为止开发的机器将Horn的猜想扩展到II_1型冯诺伊曼因子。colin和Dykema最近的一个结果可能允许人们使用他们的方法来解决cones嵌入问题，即是否每个类型的II_1因子都可以嵌入到超有限II_1因子的超幂中，这是算子代数中的一个基本问题。自伴随矩阵和的特征值问题与代数几何、交理论、表示理论和组合学有着密切的联系。拟议的项目将从操作员理论的角度提供急需的洞察力。对II_1型因子的推广将给代数几何和表示理论带来有趣的问题和反馈。PI是佐治亚理工学院科学学院ADVANCE教授。她正在与乔治亚理工学院的其他人合作，促进女性在科学和工程领域的学术进步。她还与AMS合作，研究那些特别影响女性数学家在学术界进步的问题。这个提议的项目将是她研究计划的重要组成部分，这将极大地帮助她在这方面的努力。

AbstractLiSince Klyashko's major breakthrough in the mid 1990s, there has been great excitements and renewed interests on a wide range of problems related to the Horn's conjecture in representation theory, algebraic geometry, and combinatorics. In particular, significant results were obtained by Balkale, Buch, Fulton, Knutson, Tao, Woodward, and many others.In contrast, the Horn's conjecture has not attracted as much attention in operator theory, even though the conjecture was formulated in terms of self-adjoint matrices. The proof of Horn's conjecture uses highly nontrivial tools from algebraic geometry and combinatorics.For the past ten years, it has been an ongoing project of Bercovici and the PI to find a constructive proof of Horn's conjecture which can be generalized to the von Neumann algebra setting.Recent progress has shown new promises. The propose project will provide new understanding of the eigenvalue problem and the intricate geometry of the eigenflags. The PI and her collaborators also plan to use the machinery that they have developed so far to extend the Horn's conjecture to type II_1 von Neumann factors. A recent result of Collin and Dykema may allow one to use their approach to settle the Connes'embedding problem, i.e., if every type II_1 factor can be embedded in the ultrapower of the hyperfinite II_1 factor, a fundamental question in operator algebra.The problem of eigenvalues of sums of selfadjoint matrices has intimate connections with algebraic geometry, intersection theory, representation theory, and combinatorics. The proposed project will provide the much needed insight from the operator theory point of view.The generalization to type II_1 factors will bring interesting problems and feedbacks to algebraic geometry and representation theory.The PI is the Georgia Tech ADVANCE professor in the College of Sciences. She is working with others at Georgia Tech to promote the advancement of women in science and engineering in academic.She is also working with AMS on issues that are especially affecting the advancement of women mathematicians in academics. The proposed project will be an essential part of her research program that will help her greatly at this endeavor.