Research
of Michaela Vancliff
Research
of Michaela Vancliff
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The UTA research group Noncommutative Algebraic Geometry, Representation Theory and their Interactions
UTA's College of Science magazine article about Dr. Vancliff and her research field, written by G. Pederson
For those wishing to use W. Schelter's Affine program with, or without, maxima
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 https://www.math.washington.edu/~smith/Research/research.html for a more complete list.
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: Hung Tran, Khoa Nguyen, Saber Ahmed, Ryan Jones, Jose Lozano and Lloyd Nesbitt. 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.
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 teacher at Valentines High School 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.
The main goal of Affine is to apply Bergman's Diamond Lemma to finitely generated algebras that are finitely presented (noncommutative or commutative, graded or ungraded). 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 here. Readers should note, however, that I am only a user, not a developer. I believe these comments are accurate as of Dec 2021.
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.* .
A new version of Affine is available as a loadable package in the free Maxima program. Maxima can be obtained from https://maxima.sourceforge.net/ . If one is using Fedora Linux, it is available via Fedora Linux' repository, and can be downloaded and installed using the ``dnf'' command. Similarly for some other flavors of Linux such as Arch Linux by using the appropriate commands. If the user wishes to use Affine and the vim editor together, one must download the clisp version of Maxima.
On Fedora Linux and/or RHEL, one can use Maxima with the Affine package at the command line. However, there is a more user-friendly command at the command line, namely rmaxima. This latter command allows one to reuse previous commands. To date, I have not been able to get any Maxima gui (xmaxima and wxMaxima) to work properly with all the commands defined in the Affine package. I usually use Maxima, Affine and vim (a file editor) together in a file that I edit while running Maxima (see previous “bullet” and next “bullet”).
On a Linux or Unix/Mac machine, one can use Maxima with either emacs or the vim editor. Not being familiar with emacs, I will focus here on vim and Maxima on a Fedora Linux machine. There are at least 2 ways to use Maxima and the vim editor together. One way is to type the lines of code, say lines a through b, in a file using vim, and then use the ``:'' escape to run lines a through b piped through Maxima using the ``!'' key. E.g., :a,b!/usr/bin/maxima (where the program is called by the file /usr/bin/maxima). This method appears to require Maxima to have been compiled with clisp. Note that Maxima eats the lines of code, so the code should first be typed twice in the file (or, rather, typed and then copy/pasted), with the latter copy run through Maxima, so that one can keep a copy of what is entered in Maxima. The second method is one I have not tried myself, but entails the use of vim plugins; see the website: https://github.com/baruchel/vim-notebook . In theory, these methods should also work on a Mac or Macbook as long as one is able to prevent buffering.
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.
For people wanting to use Maxima withOUT the Affine package, my preferred gui is wxMaxima (also available for free from sourceforge.net and from Fedora's repository). I use wxmaxima to help me write questions to give to students (on homework and tests), such as questions on Jordan normal form, application of Gram-Schmidt orthogonalizaton procedure, etc. (Recall that I do NOT use wxmaxima with Affine, as, to date, I have not been able to get wxMaxima to work properly with all the commands defined in the Affine package.)
Documentation on Affine & Maxima can be found here, here and here. A simple example for computing a partial Hilbert series of a graded algebra (&/or for finding a vector-space basis of an algebra) using Maxima with the Affine package is available here.
Another program that does some, but not all, of the activities of Affine is Bergman. As far as I know, Bergman (as of Aug 2019) does not compute a vector-space basis for ungraded algebras.
My talk at MSRI Feb 23, 2000: The Points of Quadratic Algebras |
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My talk at AMS meeting held at the University of Hawaii in Mar 2012: Classifying Quadratic Quantum Planes using Graded Skew Clifford Algebras |
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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 without pauses (larger file) |
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talk written formally in pdf file. |
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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 |
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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) |
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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 ℙ3s |
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My talk given at the ``Algebra Extravaganza!” in Philadelphia, July 2017: One-Dimensional Line Schemes |
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My talk (plenary lecture) "Generalizing Classical Clifford Algebras, Graded Clifford Algebras and their Associated Geometry," delivered at The 12th International Conference on Clifford Algebras and Their Applications in Mathematical Physics, Hefei, PRChina (Covid-19 => moved to virtual setting) https://www.smartchair.org/hp/ICCA2020/ |
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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, 15-22 by NSF grant DMS-0900239 and 20-28 by NSF grant
DMS-1302050.
Quadratic Algebras Associated with the Union of a Quadric and a Line in ℙ3, J. Algebra 165 No. 1 (1994), 63-90. journal article (preprint as a pdf file).
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).
Embedding a Quantum Nonsingular Quadric in a Quantum ℙ3 (with Kristel Van Rompay), J. Algebra 195 No. 1 (1997), 93-129. journal article (preprint as a pdf file).
Some Quantum ℙ3s 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).
Some Quantum ℙ3s with One Point (with Brad Shelton), Comm. Alg. 27 No. 3 (1999), 1429-1443. journal article (preprint as a pdf file).
Embedding a Quantum Rank Three Quadric in a Quantum ℙ3 (with Brad Shelton), Comm. Alg. 27 No. 6 (1999), 2877-2904. journal article (preprint as a pdf file).
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).
Four-dimensional Regular Algebras with Point Scheme a Nonsingular Quadric in ℙ3 (with Kristel Van Rompay), Comm. Alg. 28 No. 5 (2000), 2211-2242. journal article (preprint as a pdf file).
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. journal article (preprint as a pdf file)
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).
Schemes of Line Modules II (with Brad Shelton), Comm. Alg. 30 No. 5 (2002), 2535-2552. journal article (preprint as a pdf file).
Some Finite Quantum ℙ3s 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).
Constructing Clifford Quantum ℙ3s with Finitely Many Points (with Darin R. Stephenson), J. Algebra 312 (2007), 86-110. journal article (preprint as a pdf file).
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).
Classifying Quadratic Quantum ℙ2s 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).
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 Amer. Math. Soc. (2013), 241-250. journal article (preprint as a pdf file).
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).
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).
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).
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).
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)
The One-Dimensional Line Scheme of a Certain Family of Quantum ℙ3s (with Richard G. Chandler), J. Algebra 439 (2015), 316-333. journal article (preprint as a pdf file)
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)
The One-Dimensional Line Scheme of a Family of Quadratic Quantum ℙ3s (with Derek Tomlin), J. Algebra 502 (2018), 588-609. journal article (preprint as a pdf file).
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), Proc. Amer. Math. Soc. 147 No. 10 (2019), 4135-4146. journal article (original preprint as a pdf file later preprint as a pdf file)
Skew Clifford Algebras (with Thomas Cassidy), Journal of Pure and Applied Algebra 223 No. 12 (2019), 5091-5105. journal article (original preprint as a pdf file later preprint as a pdf file)
The Quantum Spaces of Certain Graded Algebras Related to 𝔰𝔩(2, 𝕜) (with Richard G. Chandler), Algebras and Representation Theory 23 (2020), 1781-1796. journal article (preprint as a pdf file)
A Geometric Invariant of 6-dimensional Subspaces of 4×4 Matrices (with Alexandru Chirvasitu and S. Paul Smith), Proc. Amer. Math. Soc. 148 (2020), 915-928. journal article (preprint as a pdf file).
Generalizing Classical Clifford Algebras, Graded Clifford Algebras and their Associated Geometry, Adv. App. Clifford Alg. 31 (2021), 12 pgs. Journal article (preprint as a pdf file)
Twisting Systems and some Quantum ℙ3s with Point Scheme a Rank-2 Quadric (with Hung V. Tran), , in ``Recent Advances in Noncommutative Algebra and Geometry,'' Eds. K. A. Brown et al, Contemporary Math., Amer. Math. Soc., to appear. (preprint as a pdf file)
An Example of a Quadratic AS-Regular Algebra without any Point Modules, in ``Recent Advances in Noncommutative Algebra and Geometry,'' Eds. K. A. Brown et al, Contemporary Math., Amer. Math. Soc., to appear. (preprint as a pdf file)
S. Paul Smith’s Contributions to Mathematics (with K. A. Brown, T. J. Hodges, R. Kanda, A. Nyman and J. J. Zhang), in ``Recent Advances in Noncommutative Algebra and Geometry,'' Eds. K. A. Brown et al, Contemporary Math., Amer. Math. Soc., to appear.
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from
https://amath.colorado.edu/appm/staff/fast/java/qs
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