Opportunities for undergraduate research.
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Curious images from research

(Some information needs updating. This web page has been moved twice, so some links are likely broken -- contact me if so.)

Curriculum Vitæ

Recent research:

• any kind of computational algebra
• Faugère's F5 algorithm for computing Gröbner bases
  
• skipping S-polynomial reduction by checking the leading terms
• collaboration with a chemist who studies gene expression
• Gröbner basis detection (on hold)
  

Sample posters:

• ECCAD 2003: Combinatorial Criteria for S-polynomial reduction
• ECCAD 2007: A sufficient and necessary criterion for Buchberger's chain condition
  (Note: Theorems 1 and 2 have a typo; should read "implies" not "iff")
• LETTERS day 2008:
• Determinants in Wonderland (with students Deanna Leggett and Ashley Sanders)

Publications:

• Combinatorial Criteria for Gröbner Bases (PhD dissertation: PDF)
  
• Are Buchberger's Criteria necessary for the chain criterion?  (joint with Hoon Hong) (Journal of Symbolic Computation, July 2007, vol. 42 pgs. 717-732, DOI 10.1016/j.jsc.2007.02.002) (accepted author's manuscript: PDF)

Preprints:

Preprints on arxiv.org have not been refereed, unless otherwise noted. My page on arxiv is here.

• An extension of Buchberger's criteria for Gröbner basis decision (arxiv), submitted
• Generalizing Dodgson's method: a "double-crossing" approach to computing determinants (joint with Deanna Leggett and Eve Torrence) (arxiv), submitted
• F5C: a variant of Faugere's F5 algorithm with reduced Gröbner bases (joint with Christian Eder) (arxiv), presented at MEGA'09, to be submitted for special issue of a journal

Samples of talks:

• A two-part introduction to what I study:
  • From Gauss to Gröbner Bases
    (an introduction to Gröbner Bases, given at the University of Southern Mississippi's Colloquium on Pure and Applied Mathematics as well as several other institutions)
  • Criteria on Leading Terms for S-polynomial Representations
    (a more detailed view of my research, originally given to the symbolic computation research group at Universität Passau)
• Understanding and Implementing F5
  (given at Universität Kaiserslautern, March 2008)
• Conference talks:
• Western Florida MAA meeting, Pensacola, FL: Determinants in Wonderland
  (joint research with two undergraduate students, Deanna Leggett and Ashley Sanders; see poster above)
• ACA 2004, Beaumont, TX: Gröbner Basis for Two Polynomials
• ACA 2005, Nara, Japan: Combinatorial Criteria for Gröbner Bases (Young Researchers' Session)
• NCSU Symbolic Computation Seminar:
• When can we skip S-polynomial Reduction?
  
• NCSU Graduate Student Algebra Seminar:
• Combinatorial Criteria for Groebner Bases

Other talks given:

• LA/MS Sectional Meeting (2007)
• Louisiana State University Algebra Seminar (2007)
• Mississippi State University Undergraduate Mathematics Seminar (2007)
• Seton Hall University Mathematics Department (2007)
• Symbolic Computation Research Group, Moscow State University, Russia (2005)
  

Notes:

• Lecture Notes in Computer Algebra at NC State University

Software:

• "Toy/demonstration" implementations of F5 based on this pseudocode and this variant (last updated January 2009)
• in Singular (joint work with Christian Eder)
• f5ex.lib
  uploaded November 2008; revised January 2009; latest version June 2009
  (example systems; you need this or the other libraries will complain)
• f5_library.lib
  uploaded May 2008; revised January 2009; latest version June 2009
  (F5, F5R, and F5C: use commands basis, basis_r, and basis_c)
• f5r_library.lib
  uploaded November 2008; obsolete (incorporated into f5_library.lib)
• f5c_library.lib
  uploaded November 2008; obsolete (incorporated into f5_library.lib)
• Changes to Till Steger's Magma implementation,
  which when implemented cause it to terminate on the non-terminating example in that implementation
  uploaded 13 June 2009
  
• in Sage
• f5.py
  uploaded November 2008; latest version June 2009 (there was a bug in the January 2009 version)
  (primarily Martin Albrecht's code)
• f5.pyx
  uploaded November 2008
  (entirely Simon King's code)
• f5_sugar.py
  uploaded November 2008
  (an experimental version based on Albrecht's code to see whether F5 and Sugar would play together; unfortunately this will remain slow unless I find a way to make Sage and/or Singular report the quotients from polynomial division, not just the remainder)
• in Maple: F5_module.mw
  uploaded July 2007; latest version 26 September 2007
  (this version is quite slow and unmaintained)
• Maple:
  • plotacceptableregions3d.mpl
    A Maplet to plot regions that satisfy the criterion described in the joint paper with Hoon Hong, above. Public domain.
    last update/big fix: 14 September 2004
    
  • GBDetection.mw
    A Maple worksheet and module implementing a criterion for detection of a Gröbner basis of two polynomials, using an algorithm from the dissertation above.
    last update/bug fix: 19 November 2007
    
  • efc_buchberger_module.mw
    An implementation of Buchberger's algorithm, using a new criterion for skipping S-polynomial reductions from a paper I have submitted for publication. The most recent version includes a comparison with the Gebauer-Möller algorithm. GPL.
    uploaded 10 July 2007; latest version 23 November 2007
    
• SmartEiffel:
(requires the 1.1 version of the SmartEiffel compiler)
• SPolys.tar.gz
  Public domain.
  last update/bug fix: 12 August 2004
  (This has seen some work and upgrading since then, but I haven't checked the new version for release. I'll do that if someone asks.)
• html_css scripts for SmartEiffel's short tool 

Associations:

• American Mathematical Society
• Mathematical Association of America
• Assocation for Computing Machinery
  (not the whole thing, actually, but SIGSAM: Special Interest Group for Symbolic and Algebraic Manipulation)
• Symbolic Solutions Group (in the past)

Past Service:

• coordinator, NCSU Graduate Algebra Seminar, Fall 2001
  

Curious images from research:

• New possibilities for a bridge term to skip S-polynomial reduction (animated GIF)
  

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Some more interesting results will be posted Real Soon Now^TM.

If you have questions or suggestions or bug reports, please do contact me.