Algebraic Groups, Arithmetic Groups and Locally Symmetric Spaces

Gopal Prasad将研究几个算术和几何问题。在最近的工作与布赖恩康拉德和奥弗加伯，他已分类伪约化群在任意领域的奇的特点，也超过算术重要的地方和全球功能领域的特点2。普拉萨德建议与康拉德一起确定特征为2的任意域上的伪约化群的分类。伪约化群的研究和分类是线性代数群理论的重要组成部分。Prasad与Andrei Rapinchuk一起引入了“算术群的弱可分性”的概念。他们已经开发出技术来研究弱可分解算术群之间的关系，这些技术已经产生了有用的和令人惊讶的强大结果。这些结果帮助他们决定了两个闭测地线的长度相等的局部对称空间何时必须彼此“可交换”，并解决了经典问题“人们能听到鼓的形状吗？“在几何学中。 Prasad和Rapinchuk将继续进一步探索他们的技术，并将其应用于解决该领域的其他问题。代数几何，微分几何和表示论是现代数学的重要和活跃领域。普拉萨德在过去为这些领域做出了重要贡献。他未来的工作，以及他可能开发的新技术，应该对这些领域的研究人员有用。普拉萨德将投入相当多的时间在未来三年内写一本研究生水平的教科书给予快速，但全面，介绍理论的李群，写一本书的Bruhat，山雀理论的用户，并写一本参考书的著名“同余子群问题”。这些书将有助于研究生和年轻的数学研究人员的教育。目前，还没有关于Bruhat-Tits理论和同余子群问题的教科书。

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.