Horn's Conjecture and related Problems

霍恩猜想及相关问题

• 批准号: 0800629
• 负责人: Wing Suet Li
• 金额: $ 15.77万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-15 至 2012-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800629&HistoricalAwards=false
• 关键词: Horn; Conjecture; related; Problems

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.

自从Klyashko在20世纪90年代中期取得重大突破以来，人们对Horn猜想在表示论、代数几何和组合数学中的广泛问题产生了极大的兴趣。特别是，显着的结果，获得了由Mrsale，布赫，富尔顿，克努森，陶，伍德沃德，和许多其他。相比之下，霍恩的猜想并没有引起太多的关注，在运营商理论，即使该猜想制定的自伴矩阵。Horn猜想的证明是利用代数几何和组合数学中非常重要的工具，Bercovici和PI在过去的十年中一直致力于寻找Horn猜想的构造性证明，并将其推广到von Neumann代数环境中，最近的研究取得了新的进展.该项目将提供新的理解的特征值问题和复杂的几何特征旗。PI和她的合作者还计划使用他们迄今为止开发的机器将霍恩猜想扩展到II_1型冯诺依曼因子。柯林和戴克玛最近的一个结果可能允许人们使用他们的方法来解决康纳斯嵌入问题，即，超有限II_1型因子的超幂是算子代数中的一个基本问题，自伴矩阵和的特征值问题与代数几何、交理论、表示论、组合学等学科有着密切的联系。拟议的项目将提供急需的洞察力，从算子理论的观点。推广到II_1型因素将带来有趣的问题和反馈代数几何和表示理论。PI是科学学院的格鲁吉亚技术进步教授。她正在与其他人在格鲁吉亚技术，以促进妇女在科学和工程学术的进步。她还与AMS的问题，特别是影响妇女数学家在学术进步。拟议的项目将是她的研究计划，这将有助于她在这奋进的努力很重要的一部分。