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**11 - 17**of**17**### Research Program

, 2012

"... 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 ong ..."

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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 Căldăraru 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 automorphisms on the symmetric algebra S(V) of a vector space V (i.e. a polynomial

### Current Research: Noncommutative Representation Theory

"... Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unify ..."

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Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an algebra, but also a coalgebra, and the notion of an action preserving the structure of a ring uses this property. The

### MODULE INVARIANTS AND BLOCKS OF FINITE GROUP SCHEMES

"... Abstract. We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the representation-theoretic constructions of E. Friedlander and J. Pevts ..."

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Abstract. We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the representation-theoretic constructions of E. Friedlander and J. Pevtsova. Let G be a finite group scheme over an algebraically closed field k of character-istic p> 0, and let k[G] denote the coordinate algebra (or representing algebra) of G. The algebra k[G] is a finite dimensional commutative Hopf algebra, and rep-resentations of G are equivalent to right co-modules for k[G], which in turn are equivalent to left modules of the “group algebra ” kG: = Homk(k[G], k). As kG is finite dimensional, it can be decomposed uniquely as an algebra into the direct product of its indecomposable two-sided ideals called the blocks of kG. Any kG-module breaks up as the direct sum of modules which lie in a block, reducing the study of kG-mod to the study of the module categories of its blocks. In the case that G is a finite group, the representation theory of a block is governed to a certain extent by its defect group, which is a particular p-subgroup

### Past Research

"... I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program has involved collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past res ..."

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I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program has involved collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past research projects, which fall loosely into three categories: Hochschild cohomology and deformations. A large part of my research program has involved 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 Căldăraru 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 automorphisms on the symmetric algebra S(V) of a vector space V (i.e. a polynomial