Intertwining Liftings and Young Tableaux Related Problems

摘要：本课题旨在更好地理解作用于复、可分离、无限维Hilbert空间上的有界线性算子的结构。首席研究员计划继续对交换子提升定理的理论进行研究，包括数值方面和对多算子设置的推广。同时，她将继续研究自伴随算子的特征值和及其与代数几何中的交点理论、Young表和表示理论的联系。这个项目是为了更好地理解作用于希尔伯特空间的线性算子理论（“算子理论”）。本提案将重点关注两个主要研究领域。一个是在换向举升定理的理论领域，它已被证明在工程控制理论中是有用的。另一个是关于（自伴随）算子和的特征值问题。这个问题起源于1912年魏尔研究偏微分方程时发表的一篇著名论文。直到最近才有了重大突破，其解依赖于代数几何中一些非常深奥的定理。我们希望这个项目能从算子理论的角度对这个问题有所了解。此外，我们希望将现有的理论扩展到一个可以应用于量子物理的环境。

Abstract: This project is to try to obtain a better understanding of the structure of bounded linear operators acting on a complex, separable, infinite dimensional Hilbert space. The principal investigator plans to continue her research on the theory of commutant lifting theorem, including the numerical aspect and the generalization to the multioperator setting. At the same time, the principal investigator will continue her research on sums of eigenvalues of self-adjoint operators, and its connections with intersection theory in algebraic geometry, Young tableaux, and representation theory. This project is to try to obtain a better understanding of the theory of linearoperators acting on Hilbert space (``operator theory''). There are two main research areas that this proposal will focus on. One is in the area of the theory of commutant lifting thorem, which has proven to be useful in control theory in engineering. The other one is on a problem about eigenvalues of sums of (self-adjoint) operators. This problem originated in a celebrated paper of Weyl in 1912 while he was studying partial differential equations. Only recently that there was a significant break through and the solution depends on some very deep theorems in algebraic geometry. It is our hope that this project will bring some insight into the problem from the operator theory point of view. Furthermore we want to extend the existing theory to a setting that can be applied to quantum physics.