Algebraic Groups, Arithmetic Groups and Locally Symmetric Spaces

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Gopal Prasad will work on several problems of arithmetic and geometric interest. In a recent work with Brian Conrad and Ofer Gabber, he has classified pseudo-reductive group over arbitrary fields of odd characteristics, and also over arithmetically important local and global function fields of characteristic 2. Prasad proposes to work with Conrad to determine the classification of pseudo-reductive groups over an arbitrary field of characteristic 2. Study and classification of pseudo-reductive groups is very important for the theory of linear algebraic groups. Together with Andrei Rapinchuk, Prasad has introduced the notation of "weak commensurability of arithmetic groups". They have developed techniques to study the relationship between weakly commensurable arithmetic group which have yielded useful and surprisingly powerful results. These results have helped them to decide when two locally symmetric space for which the sets of length of closed geodesics are equal must be "commensurable" to each other, and also solve the classical problem "can one hear the shape of a drum?" in geometry. Prasad and Rapinchuk will continue to further explore their techniques and apply them for solution of other problems in the area. Algebraic geometry, differential geometry and representation theory are important and active areas of modern mathematics. Prasad has made fundamental contributions to these areas in the past. His future work, and the new techniques he is likely to develop, should be useful for researchers in these areas. Prasad will devote considerable time in the next three years to write a graduate level text-book giving a rapid, but comprehensive, introduction to the theory of Lie groups, to write a book on the Bruhat-Tits theory for the users, and to write a reference-book on the celebrated "congruence subgroup problem". These books will help in education of graduate students and young researchers in mathematics. At present, there are no text-books on the Bruhat-Tits theory and on the congruence subgroup problem.