Research of Michaela Vancliff

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Last revision: Aug 26, 2014.

Background on my Research Area

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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.

Noncommutative Algebraic Geometry, Representation Theory and their Interactions

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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.

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 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.

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For those wishing to use W. Schelter's Affine program......

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.

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


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


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


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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


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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.


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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 )


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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)

Publications

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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.

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

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

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

  4. 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)

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

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

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

  8. 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

  9. 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

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

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

  12. 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

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

  14. Generalizations of Graded Clifford Algebras and of Complete Intersections (with Thomas Cassidy), J. London Math. Soc. 81 (2010), 91-112. official article     corrigendum

  15. Classifying Quadratic Quantum 2s by using Graded Skew Clifford Algebras (with Manizheh Nafari and Jun Zhang), J. Algebra, 346 No. 1 (2011), 152-164. official article

  16. Generalizing 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 article

  17. Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras (with Manizheh Nafari), Comm. Alg., to appear; 7 pages. (pdf file)

  18. On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings, Comm. Alg. 43 No. 2 (2015), 460-470. official article

  19. 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)

  20. Corrigendum to ``Generalizations of Graded Clifford Algebras and of Complete Intersections'' (with Thomas Cassidy), J. London Math. Soc., to appear; 7 pages. (pdf file) (doi: 10.1112/jlms/jdu030 (as of today, full text article available, but conceivably this could change once published on paper))

  21. Point Modules over Regular Graded Skew Clifford Algebras (with Padmini P. Veerapen), J. Algebra, to appear; 11 pages. (pdf file) (doi: 10.1016/j.jalgebra.2014.08.003 )



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Cool Quadrics


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

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