Arithmetic Groups

Yale University, New Haven CT

Investigators

Abstract

We plan to study arithmetic groups for their own sake and for their applications to other branches of mathematics and computer science. Two central themes are the subgroup growth of arithmetic lattices and the combinatorics of the simplicial complexes obtained by taking quotients of Bruhat-Tits buildings modulo arithmetic groups. Arithmetic groups are at the crossroads of several central areas of mathematics: Lie groups, number theory, geometry and combinatorics. Their study can be carried out using a wealth of different methods and shed light on different directions. A previous work has led to the construction of Ramanujan graphs which turn out to have remarkable applications to computer science and combinatorics. One of the goals of this research will be to study higher dimensional analogues with the hope of getting deeper applications.