CAREER: Lattice Point Distribution and Homogeneous Dynamics

这个项目将使用解析和几何工具来研究数论中的经典问题。所讨论的基本算术问题是寻找、计数和理解数学中自然出现的代数方程的整数（或有理）解的分布。这些问题的简单性和内在美，以及解决这些问题的不成比例的深度和努力，自古希腊以来就激励着他们的研究。在许多情况下，所讨论的代数方程的对称性具有丰富的几何和解析结构。利用这种结构，可以将算术问题转化为对称空间上的几何和动态问题。理解一个问题的算术特征如何在相应空间的几何中表现出来，可以在算术和几何现象之间建立联系，并促进这两个领域的知识发展。本提案的研究是由教育和推广活动补充的，包括创建一个夏季研究研讨会和一个分析数论的研究生研讨会。PI将研究齐次动力学中的两类问题，它们都源于算术。第一类问题是齐次空间上单幂流的收缩目标问题。齐次空间上可对角群作用的收缩目标问题已经得到了很好的理解。除了一些特殊的算术情况外，对幂等流（以及其他慢速混合作用）的相应问题还不清楚。PI将使用谱理论和解析数论的方法，结合齐次动力学和遍历理论的方法，来分析单幂流的收缩目标问题及其在经典度量丢番图近似领域的应用。第二类问题是关于齐次空间上闭轨道的平移分布问题。闭子群轨道的平动分布的研究是一个有许多应用的有趣问题。当轨道是紧致的或有限测度的，有许多技术可以研究其平移的极限分布，这些技术可以应用于研究代数变量中整数点分布的经典问题；它们也适用于自同构形式的l函数的分析应用。PI将把这样的结果扩展到无限大的轨道上，重点是具有有趣的算术应用的情况。

This project will use analytic and geometric tools to study classical problems in number theory. The basic arithmetic problems in question are finding, counting, and understanding the distribution of integer (or rational) solutions to algebraic equations that arise naturally in mathematics. The simplicity and intrinsic beauty of these problems, and the disproportionate depth and effort of their resolution, has inspired their study since ancient Greece. In many cases, the symmetries of the algebraic equations in question have a rich geometric and analytic structure. Using this structure it is possible to translate the arithmetic problems into geometric and dynamic problems on spaces of symmetries. Understanding how the arithmetic features of a problem manifest in the geometry of the corresponding space creates a link between arithmetic and geometric phenomena, and advances knowledge in both fields. The research in this proposal is complemented by educational and outreach activities, including the creation of a summer research workshop and a graduate seminar on analytic number theory.The PI will study two types of problems in homogenous dynamics, both originating from arithmetic. The first type of problems are shrinking target problems for unipotent flows on homogenous spaces. Shrinking target problems for diagonalizable group actions on homogenous spaces are very well understood. The corresponding problems for unipotent flows (as well as other slow mixing actions) are not yet understood, except for some special arithmetic cases. The PI will use methods from spectral theory and analytic number theory, in combination with methods of homogenous dynamics and ergodic theory, in order to analyze shrinking target problems for unipotent flows and their applications to the classical field of metric Diophantine approximations. The second type of problems regards the distribution of translates of closed orbits on homogenous spaces. The study of the distribution of translates of closed subgroup-orbits is an interesting problem with many applications. When the orbits are compact, or of finite measure, there are a number of techniques to study the limiting distribution of their translates, and these can be applied to study the classical problem of distribution of integer points in algebraic varieties; they are also amenable to more analytic applications in the study of L-functions of automorphic forms. The PI will extend such results to orbits of infinite measure, with an emphasis on cases having interesting arithmetic applications.