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.
Department Members in This Field
Faculty
- Roman Bezrukavnikov Representation Theory, Algebraic Geometry
- Alexei Borodin Integrable Probability
- Pavel Etingof Representation Theory, Quantum Groups, Noncommutative Algebra
- Victor Kac Algebra, Mathematical Physics
- Ju-Lee Kim Representation Theory of p-adic groups
- George Lusztig Group Representations, Algebraic Groups
- David Vogan Group Representations, Lie Theory
- Zhiwei Yun Representation Theory, Number Theory, Algebraic Geometry
- Wei Zhang Number Theory, Automorphic forms, Arithmetic Geometry
Instructors & Postdocs
- Joseph Berleant Representation theory, Geometric algebra
- Elijah Bodish Representation theory, Link homology
- Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry
- Robin Zhang Number Theory, Automorphic Forms, Diophantine Geometry
Researchers & Visitors
- David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence
Graduate Students*
- Anlong Chua Geometric representation theory
- Ilya Dumanski Geometric and combinatorial methods in representation theory
- Haoshuo Fu
- Marisa Gaetz Group Representations, Lie Theory
- Svetlana Gavrilova
- Mikayel Mkrtchyan
- Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems
- Hao Peng
- Andrew Riesen
- Hamilton Wan
- Zeyu Wang
- Frank Wang Geometric Representation Theory
*Only a partial list of graduate students