Cohomological periods and high rank lattices

上同调周期和高阶格

• 批准号: 1401622
• 负责人: Akshay Venkatesh
• 金额: $ 72.94万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401622&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.

正如笛卡尔所实现的那样，代数方程的解可以几何实现。该观察结果是代数与几何形状之间丰富相互作用的开始。该项目将研究数字理论的两个主题。第一个涉及算术流形的形状 - 即，某些由其数量的理论对称性定义的几何形状。 PI猜想了控制其形状的新结构的存在（从数学上讲它们的拓扑），他将更详细地调查。第二个主题与高维的晶格有关。这些是现代密码学有趣的话题；另一方面，尚无令人满意的数学理论，该项目旨在发展这样的理论。更具体地说，PI提出了一个猜想，该猜想指定了算术局部对称空间的“周期”的值 - 即，通过将同源类别与归一化差分形式配对获得的数字。这种猜想很有趣，因为它暗示了这些同源性群体与某些动机共同体学群体之间的关系。 PI将研究这种猜想，并试图为此提供证据。同意晶格，PI将特别研究以下问题：n维晶格空间的直径是多少，典型的n维晶格中的短矢量如何表现，为什么LLL Lattice降低算法算法会如此良好？一个基本的工具将是对大N的GL（N）上的自动形式的分析，并且PI还将研究有关高维限制的自动形式的相关问题。