Algebra is central to modern mathematics, focusing on the structural themes that underpin and interrelate the different branches and applications of mathematics. Some of the specific areas of expertise of members of the Algebra group are as follows.
- The recently developed and evolving theory of quantum groups, which has found applications in mathematical physics, in particular in the construction of invariants that identify the geometric nature of manifolds.
- The conversion of geometry into algebra: geometric problems like the study of spaces of configurations of points are transformed into algebraic ones, yielding explicit equations.
- The classical application of group theory to mathematical physics via Lie theory.
- Hecke algebras, which have found unexpected applications in knot theory and in the study of classical geometries such as Schubert and flag varieties, thus providing links among these previously unrelated subjects.
- The study of unitary reflection groups, which extends the well known theory of Coxeter groups and which is expected to have significant ramifications for the representation theory of groups of Lie type.
- The theory of automatic groups, connecting geometry and the study of formal languages, this in turn providing a link between algebra and theoretical computer science.
Algebraic geometry, algebraic combinatorics, group theory, quantum groups, representation theory, semigroup theory, Yangians.