Arithmetic, Algebraic, and High Dimensional Aspects of Homogeneous Dynamics

The space of lattices in the Euclidean n-dimensional space Ln has been introduced by Minkowski over a century ago and has since been discovered to be a central notion in number theory as well as computer science. Its research is a confluence of geometry, arithmetic, analysis, representation theory, dynamics and complexity theory. This project will focus on several topics regarding distribution of natural collections of points and orbits in Ln and more general arithmetic homogeneous spaces. We intend to use both spectral methods and dynamical methods to study these distribution problems, in combination with other tools such as invariant theory and combinatorics. An emphasis will be given also to questions regarding lattices in high dimensions and the global geometry of the space of lattices in this case.