9700474 Rapinchuk This award provides funding for a continuing of the congruence subgroup problem for S-arithmetic subgroups of linear algebraic groups over number fields. The principal investigator will attempt to prove the centrality of the congruence kernel for some new classes of groups using a variety of techniques such as different boundedness conditions (bounded generation and similar properties), methods of representation theory (property (T) of Kazhdan), etc. He will also study the properties of groups having finite representation type and the representation varieties of some finitely generated groups. This research falls into the general mathematical field of Number Theory. Number Theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
