Eigenvalue Inequalities, Intersection of Schubert Varieties, and Related Problems

这个项目将继续研究算子理论和算子代数中自然产生的与Horn猜想有关的各种问题，以及它们与表示论、组合学和代数几何的关系。Horn猜想最初是关于自伴矩阵和的特征值的，并与代数几何和表示论有许多重要的联系。到目前为止，已经成功地推广到无限维集，包括无限维希尔伯特空间上的紧算子和有限von Neumann代数中的元。这项工作还揭示了某些算子的不变子空间的格上的一些复杂的结构性质和代数上的性质。这个项目将有助于深入了解有限von Neumann代数的结构，并可能为表示论和代数几何带来新的方向。Horn猜想被表述为线性代数中的一个问题，它在无限维环境和von Neumann代数环境中具有自然的相似之处。解决最初的Horn猜想所涉及的工具来自代数几何、表示理论和组合学。最近的进展为继续调查人员的合作努力提供了一个令人信服的理由。这里所发展的见解也有望为代数几何和表示论带来新的发现、方向和问题。这位研究员也是佐治亚理工学院科学学院的高级教授，致力于为所有教职员工，特别是女性教职员工创造一个更有利于家庭的工作环境，并作为该学院初级女教职员工的榜样和倡导者。她计划继续她的工作，提供联系的机会，并鼓励女性数学家，特别是初级女性数学家之间的讨论。

This project will continue the study on various problems naturally arise in operator theory and operator algebra related to the Horn's Conjecture, and their relationships to representation theory, combinatorics, and algebraic geometry. The Horn's Conjecture was originally formulated for the eigenvalues of sums of self-adjoint matrices and had shown many important connections with algebraic geometry and representation theory. So far, there were successful generalizations to the infinite dimensional settings, including compact operators on infinite dimensional Hilbert spaces and elements in finite von Neumann algebras. This work also revealed some intricate structure properties on the lattices of invariant subspaces of certain operators and properties on the algebra. This project will help to gain insights in the structure of finite von Neumann algebras and may bring new directions to representation theory and algebraic geometry. Horn's Conjecture was formulated as a problem in linear algebra and it has natural analogues in the infinite dimensional setting and in the von Neumann algebra setting. The tools involved in solving the original Horn's Conjecture were coming from algebraic geometry, representation theory, and combinatorics. The recent advances make a compelling case to continue the investigator's collaborative effort. It is also hopeful that the insights developed here may result in new discoveries, directions and problems for algebraic geometry and representation theory. The investigator is also the ADVANCE Professor of College of Sciences at Georgia Tech, working towards a more family friendly working environment for all faculty, in particular for women and as a role model and advocate for junior women faculty in the college. She plans to continue her work providing networking opportunities and encouraging discussions among women mathematicians, especially junior women mathematicians.