Mathematical Sciences: Pointwise Convergence of Operators, Classification of Nonsingular Transformation

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Professor Assani's research will focus on three problems. They are connected to operator theory, ergodic theory, and harmonic analysis. I) Is it possible to extend the recent result of J. Bourgain on the return times of dynamical systems to an operator theoretic context? II) Can a spectral characterization of the approximate transitivity property introduced by Connes and Woods be obtained? III) What kinds of operators have iterates which when acting on a function converge almost everywhere? This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics" can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In addition ergodic theory makes contact with operator theory and harmonic analysis when the dynamics consists of applying a single transformation again and again.