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.
Department Members in This Field
- Roman Bezrukavnikov Representation Theory, Algebraic Geometry
- Alexei Borodin Integrable Probability
- Pavel Etingof Representation Theory, Quantum Groups, Noncommutative Algebra
- Sigurdur Helgason Geometric Analysis
- Victor Kac Algebra, Mathematical Physics
- Ju-Lee Kim Representation Theory of p-adic groups
- George Lusztig Group Representations, Algebraic Groups
- Andrei Neguț Algebraic Geometry, Representation Theory
- David Vogan Group Representations, Lie Theory
- Zhiwei Yun Geometric Representation Theory, Number Theory
Instructors & Postdocs
- Elijah Bodish Lie theory, Group Representations, Link Homology
- Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry
- Kent Vashaw Noncommutative Algebra, Representation Theory, Tensor Triangular Geometry.
Researchers & Visitors
- David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence
- Anlong Chua
- Marisa Gaetz Group Representations, Lie Theory
- Arun Kannan
- Sergei Korotkikh algebraic combinatorics, integrable probability
- Vasily Krylov Geometric Representation Theory
- Calder Morton-Ferguson Geometric representation theory
- Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems
- Hao Peng
- Oron Propp Geometric representation theory
- James Tao
- Roger Van Peski Integrable probability, algebraic combinatorics, random matrix theory
*Only a partial list of graduate students