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

阿萨尼教授的研究将集中在三个问题上。它们与算子理论、遍历理论和调和分析有关。I)是否有可能将J.Bourain最近关于动力系统返回时间的结果推广到算子论的背景下？Ii)能否得到Connes和Wood引入的近似传递性的谱特征？Iii)当作用于函数时，哪种类型的运算符具有几乎收敛于所有地方的迭代？这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在“动力学”的标题下，可以把一组抽象的变换如何作用于光滑空间的现代理论放在下面。此外，当动力学由反复应用单一变换组成时，遍历理论与算符理论和调和分析相联系。

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.