Cohomological periods and high rank lattices

上同调周期和高阶格

• 批准号: 1931087
• 负责人: Akshay Venkatesh
• 金额: $ 20.25万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-02-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1931087&HistoricalAwards=false
• 关键词: Cohomological; periods; rank; lattices

As was realized by Descartes, the solution of algebraic equations can be realized geometrically. This observation was the start of a rich interaction between algebra and geometry. This project will study two topics in number theory. The first concerns the shape of arithmetic manifolds -- i.e., certain geometries defined by their number theoretic symmetries. The PI has conjectured the existence of new structures that govern their shape (mathematically speaking their topology), which he will investigate in more detail. The second topic relates to lattices of high dimension. These are a topic of interest in modern cryptography; on the other hand, a satisfactory mathematical theory of them is not yet available, and this project aims to develop such a theory. More specifically, the PI has formulated a conjecture that specifies the values of "periods" of arithmetic locally symmetric spaces -- i.e., the numbers obtained by pairing homology classes with normalized differential forms. This conjecture is interesting because it suggests a relationship between these homology groups, and certain motivic cohomology groups. The PI will study this conjecture and attempt to give evidence for it. Concerning lattices, the PI will study in particular the following questions: What is the diameter of the space of n-dimensional lattices, how do the short vectors in a typical n-dimensional lattice behave, and why does the LLL lattice reduction algorithm behave so well? A basic tool will be the analysis of automorphic forms on GL(n) for large n, and the PI will also study related questions about automorphic forms in the high-dimensional limit.

正如笛卡尔所认识到的，代数方程的解可以通过几何方式实现。这一观察是代数和几何之间丰富相互作用的开始。该项目将研究数论中的两个主题。第一个涉及算术流形的形状——即由数论对称性定义的某些几何形状。 PI 推测存在控制其形状（从数学上讲是其拓扑结构）的新结构，他将对此进行更详细的研究。第二个主题涉及高维晶格。这些是现代密码学感兴趣的话题；另一方面，目前还没有令人满意的数学理论，本项目旨在发展这样的理论。更具体地说，PI 制定了一个猜想，指定算术局部对称空间的“周期”值，即通过将同源类与归一化微分形式配对而获得的数字。这个猜想很有趣，因为它暗示了这些同调群和某些动机上同调群之间的关系。 PI 将研究这个猜想并尝试为其提供证据。关于晶格，PI将特别研究以下问题：n维晶格的空间直径是多少，典型n维晶格中的短向量如何表现，以及为什么LLL晶格约简算法表现如此出色地？一个基本工具是对大n的GL(n)上的自同构形式进行分析，PI还将研究高维极限下自同构形式的相关问题。