My Research area is Non-commutative Algebra, Regular Algebras, and Non-commutative Algebraic Geometry (AMS classification numbers 16,14). In my thesis, I investigated the regular graded skew Clifford algebras of low global dimension under the supervision of Professor M. Vancliff. The subject of the graded skew Clifford algebras is a new topic and is considered a leap forward in studying and constructing regular algebras. I am also interested in doing research in Representation Theory, Hopf Algebras, Quantum Groups, and Lie Algebras.
M. Artin, W. Schelter, J. Tate, and M. Van den Bergh introduced the notion of non-commutative regular algebras and invented new methods in non-commutative algebraic geometry in the late 1980s to study them ([AS],[ATV1],[ATV2]). Such algebras are viewed as non-commutative analogues of polynomial rings; indeed, polynomial rings are examples of regular algebras. The main results in [AS], [ATV1], and [ATV2] are the classification of regular algebras of global dimension 3 on degree-one generators. M. Artin and W. Schelter used mainly homological methods in [AS], while M. Artin, J. Tate, and M. Van den Bergh used mainly algebro-geometric methods in [ATV1] and [ATV2]. M. Artin, J. Tate, and M. Van den Bergh also defined twists by automorphisms and they proved that regularity of algebras and GK-dimension are preserved under this twisting ([ATV2,Section 8]).
- Research Statement (pdf) (This research statement does not include the open problems that I am working. Also does not include undergraduate research projects. Written in 2010, updated in May 2016.)
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- arXiv: Link
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- Conference Presentations/Seminars: Link
- Honors, Awards, and Grants: Link