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The UTA research group Noncommutative Algebraic Geometry, Representation Theory and their Interactions

UTA's College of Science magazine article about me and my research field, written by G. Pederson

My talk at MSRI, Feb 23 2000, on point modules and line modules

My talk on graded skew Clifford algebras given at AMS meeting held at UC Riverside in Nov 2009

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. More discussion on this topic may be found in the article written by G. Pederson for the UTA COS 2013-2014 magazine.

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.

I am the Director of the UTA research group Noncommutative Algebraic Geometry, Representation Theory and their Interactions, which consists of myself, Dr. Dimitar Grantcharov (co-director) and our Ph.D. students. Currently, the Ph.D. students in the group are: Justin Ahrens, Thomas Ferguson, Richard Chandler, John Griffis, Andrew Cavaness and Derek Tomlin. The group's focus is the study of modules (representations) over an algebra studied from the viewpoint of algebraic geometry, and seeing how these 2 topics feed off each other. Many of these ideas are discussed in the AGANT Seminar organized by myself, and in the local UTA seminar, Representations and Geometry Seminar, organized by D. Grantcharov and co-organized by myself, with schedule available from here.

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 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 a (full) professor.

For further details, the reader is referred to the article written by G. Pederson for the UTA COS 2013-2014 magazine.

Having received several questions from many different parts of the world in the past 12 months (Oct 2012 – Oct 2013) regarding Affine, I thought I would post online some (hopefully) helpful comments about it. Readers should note, however, that I am only a user, not a developer. I believe these comments are accurate as of October 2013.

The original program (binary?) file from the 1990s can be obtained here. It used to be the case that one could simply download the file, enter the file's name at the command line (in linux) and it would work (i.e., no fancy installation etc). However, these days, it is rarely compatible with current-day computer architecture. Depending on the PC, it should run okay using Red Hat Enterprise Linux 5.* , but not 6.* . There also appears to be a repository of the 1990s Affine located at

*ftp://ftp.ma.utexas.edu/pub/maxima/affine.tgz**.*A new version of Affine is available as a loadable package in the free Maxima program. Maxima can be obtained from http://maxima.sourceforge.net/ . If one is using Fedora Linux, it is available via Fedora Linux' distribution site, and can be downloaded and installed using the ``yum'' command. Similarly for some other flavors of Linux such as Arch Linux using the appropriate commands. If the user wishes to use Affine and the VI editor together, one must download the Clisp version.

There are

**some differences between the 1990s version and the new version**. The 1990s version uses upper-case characters for the commands, but the new version uses lower case. The new version needs non-commutative multiplication to be defined; see the example file below for details. The new version of Affine for non-Linux use sometimes requires a ``;'' at the end of a line (even after y or n answers) to operate correctly (albeit with an error message) in places where the Linux version does not require a ``;''. The new version has problems with exponents; sometimes an expression that needs to be simplified needs to be multiplied by the user first, before entering into Affine for reduction subject to the defining relations.Documentation on Affine can be found here and here. A simple example is available here.

Another program that does some, but not all, of the activities of Affine is Bergman.

My talk at MSRI Feb 23, 2000: pdf file Video of talk

My talk on regular algebras and graded skew Clifford algebras given at various venues in 2009 & 2010: pdf file. Note the corrigendum below.

My talk on graded skew Clifford algebras given at AMS meeting held at UC Riverside in Nov 2009: pdf file. Note the corrigendum below.

My talk on classifying quadratic regular algebras of global dimension three (quantum planes) using graded skew Clifford algebras given at AMS meeting held at the University of Hawaii in Mar 2012: pdf file

My talk at MSRI Jan 25, 2013:
video of
talk pdf file with pauses
pdf file without pauses (2nd file takes up
more disk space).

Note the corrigendum
below, and the talk written formally in this pdf
file.

My talk on Defining a Notion of Noncommutative Complete
Intersection via Base-Point Modules given March 21, 2014, at the conference
*Regularity and Rigidity of Noncommutative Algebras* held at the University of
Washington: pdf file (also available from here )

My talk on Defining a Notion of Noncommutative Complete Intersection via Base-Point Modules given April 12, 2014, at AMS meeting held at Texas Technological University: pdf file (see slide 11 for an example not in previous Seattle talk)

(Return to contents
list)

Publications 4-7 were funded in part by NSF grant
DMS-9622765; 8-11 by NSF grant DMS-9996056; 12-13 by NSF grant
DMS-0200757, 13-14 by NSF grant DMS-0457022, 15-21 by NSF grant
DMS-0900239 and 18-19 by NSF grant DMS-1302050.

Quadratic Algebras Associated with the Union of a Quadric and a Line in

^{3},*J. Algebra***165**No. 1 (1994), 63-90. official articleThe Defining Relations of Quantum n x n Matrices,

*J. London Math. Soc.***52**No. 2 (1995), 255-262. official articleEmbedding a Quantum Nonsingular Quadric in a Quantum

^{3}(with Kristel Van Rompay),*J. Algebra***195**No. 1 (1997), 93-129. official articleSome Quantum

^{3}s 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

^{3}s with One Point (with Brad Shelton),*Comm. Alg.***27**No. 3 (1999), 1429-1443. official articleEmbedding a Quantum Rank Three Quadric in a Quantum

^{3}(with Brad Shelton),*Comm. Alg.***27**No. 6 (1999), 2877-2904. official articlePrimitive and Poisson Spectra of Twists of Polynomial Rings,

*Algebras and Representation Theory***2**No. 3 (1999), 269-285. official articleFour-dimensional Regular Algebras with Point Scheme a Nonsingular Quadric in

^{3}(with Kristel Van Rompay),*Comm. Alg.***28**No. 5 (2000), 2211-2242. official articleNon-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 fileSchemes of Line Modules I (with Brad Shelton),

*J. London Math. Soc.***65**No. 3 (2002), 575-590. official articleSchemes of Line Modules II (with Brad Shelton),

*Comm. Alg.***30**No. 5 (2002), 2535-2552. official articleSome Finite Quantum

^{3}s that are Infinite Modules over their Centers (with Darin R. Stephenson),*J. Algebra***297**No. 1 (2006), 208-215. official articleConstructing Clifford Quantum

^{3}s with Finitely Many Points (with Darin R. Stephenson),*J. Algebra***312**(2007), 86-110. official articleGeneralizations of Graded Clifford Algebras and of Complete Intersections (with Thomas Cassidy),

*J. London Math. Soc.***81**(2010), 91-112. official article corrigendumClassifying Quadratic Quantum

^{2}s by using Graded Skew Clifford Algebras (with Manizheh Nafari and Jun Zhang),*J. Algebra*,**346**No. 1 (2011), 152-164. official articleGeneralizing the Notion of Rank to Noncommutative Quadratic Forms (with Padmini P. Veerapen),

*in*``Noncommutative Birational Geometry, Representations and Combinatorics,'' Eds. A. Berenstein and V. Retakh,*Contemporary Math.***592**(2013), 241-250. official articleGraded Skew Clifford Algebras that are Twists of Graded Clifford Algebras (with Manizheh Nafari),

*Comm. Alg.*, to appear; 7 pages. (pdf file)On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings,

*Comm. Alg*., to appear; 12 pages. (pdf file)The Interplay of Algebra and Geometry in the Setting of Regular Algebras, to appear in journal re MSRI Spring 2013 program; 19 pages. (pdf file)

Corrigendum to ``Generalizations of Graded Clifford Algebras and of Complete Intersections'' (with Thomas Cassidy),

*J. London Math. Soc.,*to appear; 7 pages. (pdf file)Point Modules over Regular Graded Skew Clifford Algebras (with Padmini P. Veerapen), preprint (July 2013), 10 pages. (pdf file)

copied
from

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

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