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.
