Gopal Prasad's primary focus has been study of reductive Lie and algebraic groups--mostly on problems which have either geometric or number theoretic origin. He has been investigating questions about certain interesting subgroups, for example, arithmetic subgroups or other large (Zariski-dense) subgroups which arise in geometry or number theory. It would be important to determine all normal subgroups of these groups. Over interesting fields like local and global fields, there are conjectures of Kneser-Tits and Margulis-Platonov which provide a description of normal subgroups. These conjectures have been settled in the affirmative for many class of groups. However, there still remain some very interesting groups for which these conjectures are open. Prasad plans to study these groups. For arithmetic groups, the famous "congruence subgroup problem" is the question whether any normal subgroup of finite index contains a congruence subgroup. The most successful approach to settling this problem has two parts: (1) Computation of the "metapectic kernel". (2) Centrality of the congruence subgroup kernel. A very precise computation of the metaplectic kernel for all groups has been done in Prasad's joint work with M.S.Raghunathan and A.S.Rapinchuk. On the other hand, the centrality of the congruence subgroup kernel is still unknown for some important class of groups. Prasad's goal is to investigate these groups and also find a conceptually better proof of the centrality in the known cases. He will continue his collaboration with A.S. Rapinchuk on this project. Another topic on which Prasad will work on is the representation theory of reductive p-adic groups. Prasad has been interested in classification of irreducible admissible representations where his goal is to obtain a classification in terms of theory of "types". A begining in this direction, for general reductive groups, was made in his joint work with Allen Moy--the geometric techniques which they introduced in representation theory have turned out to be very useful. Prasad plans to continue his research towards classification of admissible representations.
The set of symmetries of many geometric and number theoretic objects form a group. For studying these geometric and number theoretic objects, it is important to study their groups of symmetries. Prasad has been studying problems related to these groups and their subgroups. These problems have number theoretic or geometric origins and therefore their solution will have important applications to these areas. Prasad proposes to work on the Kneser-Tits and Margulis-Platonov problems which provide conjectural description of normal subgroups; both the problems have been settled for a large class of groups but some challenging cases remain open. Prasad also proposes to continue his work on the famous congruence subgroup problem where he and his collaborators Raghunathan and Rapinchuk have made many important contributions. In another direction, Prasad proposes to work on the representation theory of reductive p-adic groups. Representations of these groups arise naturally in various contexts and their study is an important component of the Langlands program in modern number theory. The geometric methods which Prasad and Allen Moy introduced in the area have turned out to be very useful. Prasad proposes to refine these methods to give a complete classification of all admissible representations.
