I study noncommutative algebras with a focus on deformation theory.   Elements in such algebras don't commute. A simple example is the skew polynomial ring consisting of polynomials in x and y but with the relation yx = -xy instead of yx=xy.  Noncommutative algebras often arise as deformations of familiar algebraic structures.  We might consider the polynomials in x and y but with the relation yx=q xy for some indeterminant q.  We obtain different noncommutative structures by setting q to different values, and we consider this whole family of noncommutative algebras as a deformation of the commutative polynomial ring.

In my Ph.D. thesis, I study deformations of algebras that arise from the action of the symmetric group Sym_n on a polynomial ring over fields of positive characteristic.  The deformations that arise are analogs of Lusztig's graded affine Hecke algebras (used to study the representation of groups of Lie type) and of Drinfeld Hecke algebras.  Similar deformations arise in algebraic combinatorics and representation theory, often in the study of symplectic orbifolds, under the name of symplectic reflection algebras or rational Cherednik algebras.