CAREER: Lattice Point Distribution and Homogeneous Dynamics

职业：格点分布和齐次动力学

基本信息

批准号：
1651563
负责人：
Dubi Kelmer
金额：
$ 40万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2025-02-28
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1651563&HistoricalAwards=false
关键词：
CAREER Lattice Point Distribution Homogeneous

项目摘要

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.

该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查找、计算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之美，以及解决这些问题的不成比例的深度和努力，自古希腊以来一直激发着他们的研究。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。使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力学问题。了解问题的算术特征如何在相应空间的几何中体现，可以在算术和几何现象之间建立联系，并增进这两个领域的知识。该提案中的研究得到了教育和外展活动的补充，包括创建夏季研究研讨会和解析数论研究生研讨会。PI将研究齐次动力学中的两类问题，两者都源于算术。第一类问题是同质空间上单能流的收缩目标问题。同质空间上可对角化群动作的收缩目标问题是很好理解的。除了一些特殊的算术情况外，单能流（以及其他慢混合动作）的相应问题尚未被理解。 PI将使用谱论和解析数论的方法，结合齐次动力学和遍历理论的方法，来分析单能流的收缩目标问题及其在度量丢番图近似的经典领域中的应用。第二类问题涉及同质空间上闭合轨道平移的分布。封闭子群轨道平移分布的研究对于许多应用来说是一个有趣的问题。当轨道是紧致的或有限测度时，有多种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簇中整数点分布的经典问题；它们还适合在自守形式的 L 函数研究中进行更多分析应用。 PI 将把这些结果扩展到无限测量的轨道，重点是具有有趣的算术应用的情况。