Cohomological periods and high rank lattices

正如笛卡儿所认识到的那样，代数方程的解可以用几何方法来实现。这一观察是代数和几何之间丰富互动的开始。这个项目将研究数论中的两个主题。第一个问题是算术流形的形状，即由它们的数论对称性定义的某些几何形状。PI已经推测出控制它们形状的新结构的存在（从数学上讲，它们的拓扑结构），他将对此进行更详细的研究。第二个主题涉及高维格。这些都是现代密码学感兴趣的话题；另一方面，目前还没有一个令人满意的数学理论，本项目旨在建立这样一个理论。更具体地说，PI已经制定了一个猜想，该猜想指定了算术局部对称空间的“周期”值-即，通过将同调类与规范化微分形式配对获得的数。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。私家侦探将研究这一猜想，并试图为其提供证据。关于晶格，PI将特别研究以下问题：n维晶格空间的直径是多少，典型n维晶格中的短向量是如何表现的，为什么LLL晶格约简算法表现得如此之好？一个基本的工具将是分析大n时GL(n)上的自同构形式，而PI也将研究高维极限下的自同构形式的相关问题。

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.