Research Program

I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program involves collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past and ongoing research projects, which fall loosely into three categories: Hochschild cohomology and deformations. A large part of my research program involves Hochschild cohomology and deformations of algebras. Hochschild cohomology is important in deformation theory, since a deformation of an algebra is infinitesimally a Hochschild 2-cocycle, and obstructions to lifting 2-cocycles to deformations live in degree 3 cohomology. My work in deformation theory began with [9], a paper I wrote with C˘ ald˘ araru and Giaquinto. We were inspired by some examples of Vafa and Witten, which are deformations of certain skew group algebras arising from orbifolds. Specifically, these are the skew group algebras S(V) G (and twisted versions), where G is a finite group acting by graded auto-morphisms on the symmetric algebra S(V) of a vector space V (i.e. a polynomial ring in a basis of V). Later I saw how to use techniques from [9] to yield a new approach to understanding graded Hecke algebras and symplectic reflection algebras (introduced independently in different settings by Drinfeld [11], Lusztig [21], and Etingof and Ginzburg [12]), for they are simply special types of deformations of S(V) G. As first steps in this direction, I wrote [47], giving more general types of deformations (but under strong restrictions on G), and [48], generalizing results of Ram and Shepler [32] on graded Hecke algebras to a twisted case. My Ph.D. student Shakalli generalized in [33] some of the ideas in [47] to deformations of S q (V)#G, where S q (V) is a quantum symmetric algebra or polynomial ring. I further developed my ideas from [48] in work with Shepler [35], where we gave a large class of new examples of graded Hecke algebras corresponding to the complex reflection group G(r, 1, n) ∼ = Z/rZ S n , as a result of our explicit computations of cohomology for these groups. We continued our program to understand generally deformations of S(V)G, motivated by the many interesting deformations already known, and by connections to geometry and combinatorics: Shepler and I recently completed three more papers in which we study in detail the algebraic structure of the Hochschild cohomology HH * …