Eigenvalue Inequalities, Intersection of Schubert Varieties, and Related Problems

本课题将继续研究与霍恩猜想相关的算子理论和算子代数中自然产生的各种问题，以及它们与表示论、组合学和代数几何的关系。霍恩猜想最初是针对自伴矩阵之和的特征值而提出的，它与代数几何和表示论有许多重要的联系。到目前为止，已经有成功的推广到无限维的情形，包括无限维Hilbert空间上的紧算子和有限vonNeumann代数中的元素。这一工作也揭示了某些算子的不变子空间格的一些复杂结构性质和代数性质。这个项目将有助于获得有限冯诺依曼代数的结构的见解，并可能带来新的方向表示理论和代数几何。 霍恩猜想是线性代数中的一个问题，它在无限维和冯诺依曼代数中有着自然的相似之处。解决霍恩猜想的工具来自代数几何、表示论和组合学。最近的进展使一个令人信服的情况下，继续研究人员的合作努力。这也是有希望的，在这里发展的见解可能会导致新的发现，方向和代数几何和表示论的问题。 调查员也是科学学院在格鲁吉亚理工学院的先进教授，朝着一个更家庭友好的工作环境，为所有教师，特别是妇女和作为一个榜样，倡导青年女教师在学院工作。 她计划继续她的工作，提供网络的机会，并鼓励妇女数学家，特别是初级妇女数学家之间的讨论。

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.