Representation theory and harmonic analysis on p-adic groups

调查员建议研究一些相互关联的主题与代表性理论和调和分析约化群在当地的领域。有四个总的主题，其中一些被进一步划分。第一个主题是关于志村品种的不良还原以及涉及晶体和同晶体的相关事项。第二个主题涉及各种设置中的转移因子：内部形式，志村品种，扭曲转移因子的下降。第三个问题是关于模p的Shimura簇的Lefschetz公式中的边界项。 第四个主题是关于有限特征局部域上群的轨道积分的几何研究方法。 在较少的技术语言的调查建议研究一些主题，属于理论的自守形式，一个美丽的数学领域的关系，所有三个主要分支的数学：代数，分析和几何。过去几十年来一直是令人兴奋的时代，工人在这方面，其中一个亮点是必不可少的使用自守形式怀尔斯的壮观证明费马大定理，另一个是工作的Lafforgue，他获得了菲尔茨奖在最后一届国际数学家大会。未来几十年将有许多令人兴奋的发展，研究人员希望通过解决提案中提出的一些问题直接促进这些发展，并间接帮助培训这一技术要求高的领域的年轻工人。

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.