Problems in operator theory

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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.