Representation Theory
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.
Faculty
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
Kent Vashaw Noncommutative Algebra, Representation Theory, Tensor Triangular Geometry.
Researchers & Visitors
David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence
Graduate Students*
Marisa Gaetz Group Representations, Lie Theory
Sergei Korotkikh algebraic combinatorics, integrable probability
Vasily Krylov Geometric Representation Theory
Calder Morton-Ferguson Geometric representation theory
Oron Propp Geometric representation theory
Roger Van Peski Integrable probability, algebraic combinatorics, random matrix theory
*Only a partial list of graduate students