CAREER: Noncommutative Analysis

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This research project centers on noncommutativity in various problems of analysis and its applications to mathematical physics. An operation * on the elements of a set is said to be commutative if x*y is equal to y*x for all elements x and y in the set. For example, the usual operations of addition and multiplication on the set of real numbers are commutative, whereas the operation of division is not commutative since, for instance, 2 divided by 3 is not equal to 3 divided by 2. Noncommutativity is an inherent phenomenon in many areas of modern mathematics and its applications. It has been a source of intricate obstacles as well as mathematical depth and beauty since J. von Neumann's formulation of quantum mechanics. The integrated program of research and education in this project is aimed at treating noncommutativity effects that arise in various problems of analysis and its applications to noncommutative geometry, mathematical physics, and operator algebras, as well as to strengthening connections between these areas of mathematics. Research and career development of students will be facilitated by workshops targeted at graduate students and junior researchers, organized in conjunction with research conferences for the broader mathematical community. University students will participate in an outreach program to enhance mathematical skills and knowledge of high school students and teachers and raise appreciation of mathematics by a general community. The research projects are vertically integrated to provide ample opportunity for student research at different levels.This research project includes establishing properties of single and multivariate operator functions that are similar to classical properties of the respective scalar functions, deriving bounds for related Schur multipliers and similar transformations, describing spectral characteristics of perturbations of differential operators, understanding geometry of higher order perturbations and finding its meaning in noncommutative geometry, and describing structure of elements in operator algebras. Resolution of these problems depends on development of innovative operator theoretic methods that are expected to emerge from a synthesis of various techniques in analysis and operator theory. The project aims to ultimately benefit areas of mathematics beyond the topics investigated in the project.