Algebraic Groups, Arithmetic Groups and Locally Symmetric Spaces

Gopal Prasad将致力于解决几个算术和几何兴趣问题。在最近与Brian Conrad和Ofer Gabber的一项工作中，他对特征为2的任意域上的伪约群以及特征为2的算术重要的局部和整体函数域上的伪约群进行了分类。Prasad建议与Conrad合作来确定特征为2的任意域上的伪约群的分类。伪约群的研究和分类对线性代数群理论是非常重要的。Prasad与Andrei Rapinchuk一起引入了“算术群的弱可公度性”这一概念。他们开发了一些技术来研究弱可公度算术群之间的关系，这些技术产生了有用的和令人惊讶的强大结果。这些结果帮助他们确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“可公度”，并解决了经典问题“人们能听到鼓的形状吗？”在几何学上。Prasad和Rapinchuk将继续进一步探索他们的技术，并将他们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的领域。普拉萨德过去在这些领域做出了根本性的贡献。他未来的工作，以及他可能开发的新技术，应该会对这些领域的研究人员有用。在接下来的三年里，普拉萨德将花相当多的时间写一本研究生水平的教科书，对李群理论进行快速而全面的介绍，为用户写一本关于Bruhat-Tits理论的书，并写一本关于著名的“同余子群问题”的参考书。这些书籍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.