Algebra and Combinatorics
Algebra and combinatorics are core areas of mathematics which find broad applications in the sciences and in other mathematical fields. Algebra is the study of algebraic structures, for example, groups, rings, modules, fields, vector spaces, and lattices. Combinatorics is the study of natural structures on discrete (often finite) sets.
Research areas in algebra include the structure and representations of Lie algebras, quantum groups, algebraic groups, toroidal algebras, Leibniz algebras, vertex algebras, and their applications in other areas of mathematics and physics. Research areas in combinatorics include algebraic combinatorics, enumerative combinatorics, geometric combinatorics, topological combinatorics, and their applications. Many faculty in the algebra and combinatorics group do algebra with a combinatorial flavor, or combinatorics with algebraic motivation or using algebraic methods.
Mathematical physics, Lie algebras, vertex algebras, integrable systems.
Topological algebra, ring theory.
Algebraic and topological combinatorics.
Representation theory, quantum groups, infinite-dimensional Lie algebras and groups, vertex algebras, algebraic combinatorics, quantum computation.
Tensor categories, mathematical physics, operator algebras, higher categories.
Algebraic combinatorics, connections to geometry and representation theory.
Representations of Lie algebras, quantum groups, vertex operator algebras; applications in number theory, combinatorics, and statistical mechanics.
Algebraic and geometric combinatorics, especially Coxeter groups, cluster algebras, and lattice-theoretic approaches.
Lie algebras, group theory, cryptography.
Distinguished Professor, Director of Graduate Programs
Algebraic statistics, computational and combinatorial algebra, mathematical phylogenetics
Real algebraic geometry, matrix theory, combinatorics, optimization