Mathematical Sciences: Multiple Recurrence Nonconventional Ergodic Averages Wiener-Wintner Return Times Theorems

Assani将研究弗斯滕伯格引入的多次重复和非常规平均问题。他将研究这些非常规平均值的逐点收敛和范数收敛。他将学习Wiener-Wintner版本的返回次数定理，Furstenberg在他的Szmeredi关于算术级数的定理的证明中使用了这些定理。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。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类的过程中与几何相联系。

Assani will investigate problems of multiple recurrence and nonconventional averages introduced by Furstenberg. He will study pointwise and norm convergence of these non-conventional averages. He will study the Wiener-Wintner versions of return times theorem which were used by Furstenberg in his proof of Szemeredi's theorem on arithmetic progressions. 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 this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.