Discrete subgroups of semisimple Lie groups

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The PI will continue studying discrete linear groups (specially arithmetic groups) and their interactions with various branches of mathematics, e.g. number theory, homogeneous dynamics and combinatorics. The focus of this project is to expand our understanding of such groups in two different directions: going from finite covolume to infinite covolume (and beyond); going from characteristic zero to positive characteristic. The PI plans to see in what extent the analytical behavior of the congruence quotients of a finitely generated group is dictated by its Zariski-topology. The PI's second goal is to study homogeneous dynamics over a local field of positive characteristic. The analytical properties of the congruence quotients of linear groups have been showed to be extremely useful in various parts of mathematics and computer science. In the past decade they have been used in affine sieve, variation of Galois representations, hyperbolic geometry and group theory. It is clear that extending these results would have immediate impacts in other branches of mathematics. As the second component of this project, the PI plans to work toward the proof of Raghunathan's conjectures for semisimple groups over a local field of positive characteristic. Many mathematicians worked on these conjectures, e.g. Dani, Margulis, Shah, Tomanov, and finally in a series of papers, Ratner completely proved these conjectures over a local field of characteristic zero. As Ratner's results have been extremely fruitful in various parts of mathematics, it is expected that any partial result toward their positive characteristic analogue would have immediate applications.One of the main tools to study an object or a structure is to understand its symmetries. That is the intrinsic reason why group theory is in a close connection with other branches of mathematics and physics. For instance the PI's work on linear groups can give us the precise algebraic conditions to construct explicit families of sparse highly connected graphs known as expanders. Expanders are extremely useful in communication, theoretical computer science (e.g. error correcting codes) and various branches of mathematics. The PI studies also the dynamical systems of algebraic nature and their deep and fruitful connections with other branches of mathematics, e.g. number theory.