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

阿萨尼将调查多次复发的问题， Furstenberg提出的非常规平均数。 他将 研究了这些非常规算法点态收敛性和范数收敛性 平均值。 他将研究维纳-温特纳版本的回归 Furstenberg在他的证明中使用了 Szemeredi的算术级数定理。 该项目涉及遍历理论的研究。 遍历 理论一般涉及理解的平均行为， 系统的动力学过于复杂或混乱， 在微观细节上进行跟踪。 在“动态”标题下， 现代理论认为， 变换作用于光滑空间。 这样遍历理论 与几何学接触，以寻求对 齐性空间

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.