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

Assani将研究由Furstenberg引入的多重递归和非常规平均问题。他将研究这些非常规平均值的点和范数收敛性。他将研究维纳-温特纳版本的返回时间定理，这是弗斯滕伯格在证明Szemeredi的等速数列定理时使用的。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。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时与几何联系在一起。

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.