Research of Michaela Vancliff

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Background on Dr. Vancliff's Research Area

Prof. Vancliff works 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. Vancliff's 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. Vancliff's publication list is below and so are some of her talks. 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.

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Noncommutative Algebraic Geometry, Representation Theory and their Interactions

Prof. Vancliff is the Director of the UTA research group Noncommutative Algebraic Geometry, Representation Theory and their Interactions, which consists of Dr. Vancliff, Dr. Dimitar Grantcharov (co-director) and various Ph.D. students. Currently, the Ph.D. students in the group are: Anthony Mastriania, Hung, Tran, Ian Lim, Dwight Williams, Khoa Nguyen, Tyler Anway and Rebekah Aduddell. 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 Dr. Vancliff, and in the local UTA seminar, Representations and Geometry Seminar, organized by Dr. Grantcharov and co-organized by Dr. Vancliff, with schedule available from here.

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Brief Biography

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

After graduating from Warwick, Vancliff was a high school teacher in greater London for one academic year, after which she joined the Ph.D. program at the University of Washington. After earning her Ph.D., she 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, she began working in the Mathematics Department of the University of Texas at Arlington in Arlington, Texas, where she is 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 during the 12-month window 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 June 2016.

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Selected Talks

My talk at MSRI Feb 23, 2000: The Points of Quadratic Algebras

pdf file

Video of talk

My talk at AMS meeting held at the University of Hawaii in Mar 2012: Classifying Quadratic Quantum Planes using Graded Skew Clifford Algebras

pdf file


My talk at MSRI Jan 25, 2013: The Interplay of Algebra and Geometry in the Setting of AS-regular Algebras (note the corrigendum below)


pdf file with pauses

pdf file without pauses (larger file)

video of talk

talk written formally in pdf file.

My talk at the conference Regularity and Rigidity of Noncommutative Algebras held at the University of Washington in March 2014: Defining a Notion of Noncommutative Complete Intersection via Base-Point Modules

pdf file   (also available from here )


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

pdf file  


My 5-minute talk given at the Fields Institute, Toronto, Canada, July 2015: The One-Dimensional Line Schemes of Two Families of Potentially-Generic Quadratic Quantum 3s

pdf file with pauses

pdf file without pauses

audio

My talk given at the ``Algebra Extravaganza!” in Philadelphia, July 2017: One-Dimensional Line Schemes


pdf file without pauses



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Publications


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, 15-22 by NSF grant DMS-0900239 and 20-28 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. journal article (preprint as a pdf file).

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

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

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

  5. Some Quantum 3s with One Point (with Brad Shelton), Comm. Alg. 27 No. 3 (1999), 1429-1443. journal article (preprint as a pdf file).

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

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

  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. journal article (preprint as a pdf file).

  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. preprint as a pdf file

  10. Schemes of Line Modules I (with Brad Shelton), J. London Math. Soc. 65 No. 3 (2002), 575-590. journal article (preprint as a pdf file).

  11. Schemes of Line Modules II (with Brad Shelton), Comm. Alg. 30 No. 5 (2002), 2535-2552. journal article (preprint as a pdf file).

  12. Some Finite Quantum 3s that are Infinite Modules over their Centers (with Darin R. Stephenson), J. Algebra 297 No. 1 (2006), 208-215. journal article (preprint as a pdf file).

  13. Constructing Clifford Quantum 3s with Finitely Many Points (with Darin R. Stephenson), J. Algebra 312 (2007), 86-110. journal article (preprint as a pdf file).

  14. Generalizations of Graded Clifford Algebras and of Complete Intersections (with Thomas Cassidy), J. London Math. Soc. 81 (2010), 91-112. journal article     corrigendum (preprint as a pdf file).

  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. journal article (preprint as a pdf file).

  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. journal article (preprint as a pdf file).

  17. Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras (with Manizheh Nafari), Comm. Alg. 43 No. 2 (2015), 719-725. (journal article) (preprint as a pdf file).

  18. Point Modules over Regular Graded Skew Clifford Algebras (with Padmini P. Veerapen), J. Algebra 420 (2014), 54-64. journal article (preprint as a pdf file).

  19. Corrigendum to ``Generalizations of Graded Clifford Algebras and of Complete Intersections'' (with Thomas Cassidy), J. London Math. Soc., 90 No. 2 (2014), 631-636. journal article (preprint as a pdf file).

  20. On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings, Comm. Alg. 43 No. 2 (2015), 460-470. journal article (preprint as a pdf file).

  21. The Interplay of Algebra and Geometry in the Setting of Regular Algebras, in ``Commutative Algebra and Noncommutative Algebraic Geometry,'' MSRI Publications 67 (2015), 371-390. journal article (preprint as a pdf file)

  22. The One-Dimensional Line Scheme of a Certain Family of Quantum 3s (with Richard G. Chandler), J. Algebra 439 (2015), 316-333. journal article   (preprint as a pdf file)

  23. A Generalization of the Matrix Transpose Map and its Relationship to the Twist of the Polynomial Ring by an Automorphism (with Andrew McGinnis), Involve 10 No. 1, (2017), 43-50.   journal article  (preprint as a pdf file)

  24. The One-Dimensional Line Scheme of a Family of Quadratic Quantum 3s (with Derek Tomlin), J. Algebra 502 (2018), 588-609. journal article   (preprint as a pdf file).

  25. The Quantum Spaces of Certain Graded Algebras Related to 𝔰𝔩(2, 𝕜) (with Richard G. Chandler), preprint 2017. pdf file

  26. A Geometric Invariant of 6-dimensional Subspaces of 4×4 Matrices (with Alexandru Chirvasitu and S. Paul Smith), Proc. Amer. Math. Soc, to appear. pdf file

  27. Associating Geometry to the Lie Superalgebra 𝔰𝔩(1|1) and to the Color Lie Algebra 𝔰𝔩2c(𝕜) (with Susan J. Sierra, Spela Spenko, Padmini Veerapen, Emilie Wiesner), preprint 2018. pdf file

  28. Skew Clifford Algebras (with Thomas Cassidy), preprint 2018. pdf file


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Cool Quadrics


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

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