Symmetries occur throughout mathematics and science. Representation theory seeks to understand all the possible ways that an abstract collection of symmetries can arise. Nineteenth-century representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. In number theory, p-adic representation theory is central the Langlands program, a family of conjectures that have guided a large part of number theory for the past forty years.
One fundamental problem involves describing all the irreducible unitary representations of each Lie group, the continuous symmetries of a finite-dimensional geometry. Doing so corresponds to identifying all finite-dimensional symmetries of a quantum-mechanical system. We've made great progress on this important problem, including work by MIT's strong faculty in this area. The representation theory of infinite-dimensional groups and supergroups is vital to string theory, statistical mechanics, integrable systems, tomography, and many other areas of mathematics and its applications.
Research interests of this group include vertex algebras, quantum groups, infinite-dimensional Lie algebras, representations of real and p-adic groups, Hecke algebras and symmetric spaces.