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.