Representation theory and harmonic analysis on p-adic groups

点击翻译按钮获取中文摘要

The investigator proposes to study a number of interrelated topics having to do with representation theory and harmonic analysis for reductive groups over local fields. There are four general topics, some of which are divided up further. The first topic concerns bad reduction of Shimura varieties and related matters involving crystals and isocrystals. The second topic concerns transfer factors in various settings: inner forms, Shimura varieties, descent for twisted transfer factors. The third topic concerns the boundary terms in the Lefschetz formula for Shimura varieties modulo~$p$. The fourth topic concerns geometric methods of studying orbital integrals for groups over local fields of finite characteristic. In less technical language the investigator proposes to study a number of topics that belong to the theory of automorphic forms, a beautiful area of mathematics with ties to all three main branches of mathematics: algebra, analysis and geometry. The last several decades have been exciting times for workers in this area, one of the highlights being the essential use of automorphic forms in Wiles's spectacular proof of Fermat's Last Theorem, another being the work of Lafforgue for which he received a Fields Medal at the last International Congress of Mathematicians. The coming decades promise many further exciting developments, and the investigator hopes to contribute to these directly, by solving some of the questions raised in the proposal, as well as indirectly, by helping to train young workers in this technically demanding field.