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.
In my NSF-project I had proposed to work on three important problems on three different areas of mathematics. The first of these problems was study of arithmetic groups. These groups appear both in number theory (the branch of mathematics which studies properties of integers and their various generalizations), and quite interestingly in modern geometry. The question which I was interested in investigating came from geometry of certain geometric figures (called locally symmetric spaces, due to their symmetric shape) and was concerned with looking at the lengths of closed locally distance minimizing paths (i.e., paths in which given two points x and y which are not too far apart, the distance between them in the geometric figure is same as the length of the segment of the path from x to y) . The question is how much the collection of these lengths differ for two diffrent geometric figures. In collaboration with Andrei Rapinchuk (of University of Virginia) I was able to answer this question which resulted in our writing a research paper. I have been collaborating with Andrei Rapinchuk for over twenty years, and we have worked on several different problems ranging from algebra to number theory and geometry. The second problem on which I worked (in collaboration with Sai-Kee Yeung, of Purdue University) is in algebraic geometry in which the geometric objects which we study are defined by polynomial equations-just in the same as in High School geometry course, ellipses, parabola and hyperbola are described. The question we studied was to determine all geometric figures of certain kind which have same Betti numbers (the n-th Betti number in lay person's term is "number of holes of dimension n") as a more familiar companion geometric figure known as the compact dual. We have shown that there are only a small number of such geometric figures and found all of them them in dimension 2 and 4. In dimension 2 there are 100 of them, but only 3 were known before. In dimension 4 none were known before our work. The third problem in the project was to classify "pseudo-reductive groups". These groups are also defined in terms of polynomial equations just like ellipses, parabolas and hyperbolas, but interestingly, there is a way to multiply any two points on these objects. Now pseudo-reductive groups arise in mathematics in a natural way and so it is of interest to study them, and classify them. Note that to "classify" in mathematics means to find a convenient way to list them all. In a research monograph written by me, jointly with Brian Conrad (of stanford University) and Ofer Gabber (IHES, France), on pseudo-reductive groups we developped a general theory of pseudo-reductive groups. We also attempted to give a classification (i.e., a complete listing) of these groups. But this turned out to be a difficult problem, and as we already knew at that time, in some cases our list was incomplete (for the more knowledgeable reader, the difficulty is due to prime number 2). In collaboration with Brian Conrad, now a complete classification of pseudo-reductive groups has been obtained and a paper on this topic has been prepared.
