Mathematical Sciences: Analysis on Semisimple Groups

首席研究员计划致力于数学分析中有关(半单)李群表示理论的问题。后者的一个合适的例子是球体的旋转群。这样的群很重要，因为它们以对称群的形式出现在数学的许多领域(例如，几何、微分方程、代数数论、数学物理)。表象理论允许人们在解决问题时利用对称性。表示理论的基石是不可约表示，特别是不可约酉表示。主要研究人员将致力于对半单李群的这些表示进行分类。以前的工作提供了相当多的例子和技术来解决这个问题，在指定的一般性程度上，这个问题非常困难，而且绝对是本主题的核心。出现了一些共同的主题，也出现了一些微妙的批评案例。计划是追求共同的主题，并集中于关键案件的概括所起的作用。

The principal investigator plans to work on problems in mathematical analysis concerning the representation theory of (semisimple) Lie groups. A suitable example of the latter is the group of rotations of a sphere. Groups like this are important because they occur in many areas of mathematics ( e.g. geometry, differential equations,algebraic number theory, mathematical physics ) as groups of symmetries. Representation theory allows one to take advantage of symmetries in solving problems. The building blocks for representation theory are the irreducible representations, particularly the irreducible unitary representations. The principal investigator will work toward classifying these representations for semisimple Lie groups. Prior work provides a considerable store of examples and techniques for attacking this problem, which in the indicated degree of generality is very hard, and absolutely central to the subject. Some common themes have emerged and some subtle critical cases have appeared. The plan is to pursue the common themes and concentrate on the role played by generalizations of the critical cases.