Problems in operator theory

该项目解决非交换分析、数学物理、非交换几何和算子代数框架中的算子理论中的开放性问题。其中一个目标是推进最近与D. Potapov和F. Sukochev共同获得的关于多算子积分和谱移函数的有力结果。新的非交换分析技术将被开发并应用于上述领域中出现的各种微扰理论问题。另一个目标是在理解与冯·诺依曼代数相关的重要算子的结构方面取得进展，继续与K. Dykema共同开始的研究。这项拟议的研究预计将加强几个数学领域之间的联系，并与现代物理学研究建立联系。许多项目对研究生和本科生都开放。研究者建议在数学的各个领域为通识教育做出贡献，并让学生参与具体的项目。拟议的研究结果将通过出版物和讲座向全世界科学界传播。这将包括写说明性文章和向学生发表演讲。研究者将继续组织会议，并在青少年和代表性不足的群体中普及数学事业。

The project addresses open problems in noncommutative analysis, mathematical physics, noncommutative geometry, and operator theory in the operator algebras framework. One of the objectives is to advance recent powerful results on multiple operator integrals and spectral shift functions that have been obtained jointly with D. Potapov and F. Sukochev. New noncommutative analysis techniques will be developed and applied to various problems of perturbation theory that have arisen in the areas mentioned above. Another objective is to make progress in understanding structure of important operators affiliated with von Neumann algebras, continuing investigation started jointly with K. Dykema.The proposed research is expected to strengthen connections between several areas of mathematics and to make a link to modern studies in physics. Many of the projects are accessible to students, at both graduate and undergraduate levels. The investigator proposes to contribute to general education in the respective areas of mathematics as well as involve students into specific projects. The results of the proposed research will be disseminated to the worldwide scientific community through publications and talks. This will include writing expository articles and delivering talks to students. The investigator will continue organizing conferences and popularizing careers in mathematics among youth and underrepresented groups.