Many geometric and arithmetic objects have symmetries that can be encoded in the mathematical notion of a group. Often such groups possess additional structure, forming what is known as an algebraic group. The PI plans to work on several research projects related to algebraic groups and their internal structure, as well as their associated geometric and arithmetic objects, especially symmetric spaces. In addition the PI plans to write three books during the duration of this award.
Recent research of the PI includes work with Brian Conrad and Ofer Gabber on the classification of pseudo-reductive algebraic groups, with Andrei Rapinchuk on arithmetically defined locally symmetric spaces, and with Sai-Kee Yeung in complex algebraic geometry. Under this award the PI proposes to investigate new directions related to these collaborations. He will also continue his work on the congruence subgroup problem and the Margulis-Platonov conjecture (the latter gives a description of the normal subgroups of the group of rational points of a simply-connected semi-simple group defined over a global field). In another direction, the PI has associated a Levi subgroup to any irreducible admissible representation of a reductive p-adic group. He proposes to investigate the role of this Levi subgroup in the construction and classification of representations. The three books the PI plans to write are one on the Bruhat-Tits theory of reductive groups over nonarchimedean local fields, one on Lie groups based on his graduate courses at the University of Michigan, and one, jointly with Andrei Rapinchuk, on the congruence subgroup problem.
