Representation theory, originating in the work of mathematicians such as Issai Schur and Ferdinand Frobenius at the end of the 19th century, studies how the structure of algebraic objects (such as finite groups or Lie algebras) can be captured by simpler, linear objects, in particular, matrices. A theme common to all of the goals of this project is the use of representation theory to study geometric spaces. Three dimensional topological quantum field theories, quantum link invariants, and vertex operators were all developed in the 1980s and 1990s using various algebraic inputs. One main objective of this project is to lift these constructions to higher dimensions through the use of algebraic structures from representation theory. This objective is part of a program known as categorification.
The program of categorification was introduced by Louis Crane and Igor Frenkel. Generally speaking, the aim is that vector spaces should be replaced by categories and actions on vector spaces should be replaced by actions of functors on categories. The categories involved in this project are of a representation theoretic or combinatorial nature. New category-theoretic ideas need to be developed in order to achieve the most ambitious goals of the categorification program. Another part of this project is the application of representation-theoretic techniques to the classification problem of certain algebraic varieties equipped with a group action. The four areas of this project are the following: (1) categorification of the Reshetikhin-Turaev invariant, (2) categorification of the Turaev-Viro invariant, (3) categorification of Lie superalgebras, and (4) understanding the multiplicative structure of the coordinate rings of affine spherical varieties.
