Multiple Pointwise Ergodic Theorems

遍历理论从统计的角度研究动力系统的长期行为。它是一个庞大而迅速发展的数学领域，对加法组合学、数论和傅立叶分析等领域的发展产生了深远的影响。这个项目的主要重点是理解自然出现在分析和遍历理论的界面上的多个多项式遍历平均的范数和逐点收敛现象。还将研究添加剂组合学中的广泛一类问题。在整个研究计划期间，该项目将有助于培养本科生和研究生以及博士后研究员。 PI还将促进数学向更广泛的社区，并鼓励代表性不足的群体的参与。点态遍历理论的主要开放问题之一是Furstenberg-Bergelson-Leibman猜想，它断言多个多项式遍历平均值几乎处处点态收敛。PI将开发傅立叶分析和加法组合学的工具，允许研究对应于不同次数多项式的多个多项式遍历平均值的逐点收敛。此外，PI将调查多项式Szemeredi的类型定理在拓扑领域和多维整数网格。 该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ergodic theory studies long-term behavior in dynamical systems from a statistical point of view. It is a large and rapidly developing area of mathematics, which has a profound impact on the development of additive combinatorics, number theory, and Fourier analysis, among other fields. The primary focus of this project is on understanding phenomena of norm and pointwise convergence for multiple polynomial ergodic averages that naturally arise at the interface of analysis and ergodic theory. A broad class of questions in additive combinatorics will also be investigated. Throughout the duration of the research program, the project will contribute to the training of undergraduate and graduate students and of postdoctoral fellows. The PI will also promote mathematics to the broader community and encourage the participation of underrepresented groups.One of the major open problems in pointwise ergodic theory is the Furstenberg-Bergelson-Leibman conjecture, which asserts that the multiple polynomial ergodic averages converge pointwise almost everywhere. The PI will develop tools in Fourier analysis and additive combinatorics allowing to study pointwise convergence for multiple polynomial ergodic averages corresponding to polynomials with distinct degrees. In addition, the PI will investigate polynomial Szemeredi's type theorems in topological fields and multidimensional integer grids. At the heart of these investigations are the so-called inverse theorems from higher order Fourier analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.