Background on my Research Area

I work in the subject of non-commutative algebra. Broadly speaking, this subject is about solving systems of "polynomial" equations where the solutions are functions (typically differential operators or matrices, etc). This means that we cannot assume that the variables in the equations commute with each other. Such equations arise in the theory of quantum mechanics, statistical mechanics, physics, etc.

The problem of solving a system of equations in non-commutative algebra may be translated to one involving an algebra over a field, and the representation theory (or module theory) of that algebra. My research is in the subarea of non-commutative algebraic geometry, which is about using geometric methods to understand the algebra and its representation theory that arise in this way.

The originators of this kind of non-commutative algebraic geometry are Michael Artin, John Tate and Michel Van den Bergh through work they did in the late 1980's. The subject has grown through the work of these people and of S. Paul Smith, Toby Stafford, Thierry Levasseur, Lieven Le Bruyn and James Zhang to name a few. New ideas and theories are continually being presented, and the research in this subject has grown considerably since the late 1980's. My publication list is below. Click on the preceding names to find other publications or go to http://www.math.washington.edu/~smith/Research/research.html for a more complete list.

Brief Biography

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I obtained my Mathematics Ph.D. in 1993 from the University of Washington (Mathematics) under the supervision of S. Paul Smith. The University of Washington is in Seattle, WA, U.S.A.
I obtained my Mathematics bachelor degree in 1986 from the University of Warwick (Mathematics), which is in the Midlands in England.
I spent 6 months of my last academic year of my PhD in the Department of Mathematics of the University of Auckland, in Auckland, New Zealand.

After graduating from Warwick, I was a high school teacher in greater London for one academic year, after which I began my Ph.D.
After getting my Ph.D, I worked for 2 years at the University of Southern California (Mathematics) in Los Angeles, CA, U.S.A.; and then for one year at the University of Antwerp (U.I.A.) in Antwerp, Belgium; and then for 2 years at the University of Oregon ( Mathematics ) in Eugene, OR, U.S.A. In August 1998, I began working in the Mathematics Department of the University of Texas at Arlington in Arlington, Texas, where I am now an associate professor.

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Talk at MSRI Feb 23, 2000: The Points of Quadratic Algebras. pdf file dvi file Video of talk

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Talk at Central Texas Algebra Conference at Baylor University, Feb 22, 2003:
The Points and Lines of Quadratic Algebras. pdf file dvi file

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Talk in workshop ``Interactions Between Noncommutative Algebra and Algebraic Geometry'', BIRS, Banff, Canada, Sept 14, 2005: Using an Algebro-Geometric Method to Construct Clifford Quantum ³s with a Predetermined Finite Point Scheme. pdf file dvi file

Publications

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The 4th-7th publications were funded in part by NSF grant DMS-9622765; the 8th-11th publications were funded in part by NSF grant DMS-9996056; the 12th and 13th publications were funded in part by NSF grant DMS-0200757, and the 13th publication was funded in part by NSF grant DMS-0457022.

Quadratic Algebras Associated with the Union of a Quadric and a Line in ³, J. Algebra 165 No. 1 (1994), 63-90. official article

The Defining Relations of Quantum n x n Matrices, J. London Math. Soc. 52 No. 2 (1995), 255-262. pdf file

Embedding a Quantum Nonsingular Quadric in a Quantum ³ (with Kristel Van Rompay), J. Algebra 195 No. 1 (1997), 93-129. official article

Some Quantum ³s with Finitely Many Points (with Kristel Van Rompay and Luc Willaert), Comm. Alg. 26 No. 4 (1998), 1193-1208. official article (title incorrect on that website)

Some Quantum ³s with One Point (with Brad Shelton), Comm. Alg. 27 No. 3 (1999), 1429-1443. official article

Embedding a Quantum Rank Three Quadric in a Quantum ³ (with Brad Shelton), Comm. Alg. 27 No. 6 (1999), 2877-2904. official article

Primitive and Poisson Spectra of Twists of Polynomial Rings, Algebras and Representation Theory 2 No. 3 (1999), 269-285. pdf file

Four-dimensional Regular Algebras with Point Scheme a Nonsingular Quadric in ³ (with Kristel Van Rompay), Comm. Alg. 28 No. 5 (2000), 2211-2242. official article

Non-commutative Spaces for Graded Quantum Groups and Graded Clifford Algebras, Clifford Algebras and their Applications in Mathematical Physics 1 (Ixtapa-Zihuatanejo, 1999), 303-320, Progress in Physics, 18, Birkhaeuser Boston, Boston, MA, 2000. pdf file

Schemes of Line Modules I (with Brad Shelton), J. London Math. Soc. 65 No. 3 (2002), 575-590. official article

Schemes of Line Modules II (with Brad Shelton), Comm. Alg. 30 No. 5 (2002), 2535-2552. official article

Some Finite Quantum ³s that are Infinite Modules over their Centers (with Darin R. Stephenson), J. Algebra 297 No. 1 (2006), 208-215. official article

Constructing Clifford Quantum ³s with Finitely Many Points (with Darin R. Stephenson), J. Algebra 312 (2007), 86-110. official article

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Cool Quadrics

copied from
http://amath.colorado.edu/appm/staff/fast/java/qs

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