Math 648: Computational Algebraic Geometry

Instructor: Frank Sottile
Lectures: TΘ 14:20–15:35 Blocker 506A
Course webpage: www.math.tamu.edu/~sottile/teaching/19.1/648.html
Grading: Some homework/computer projects and end-of-term projects.
Prerequisites: Graduate algebra or permission of instructor. This may be taken concurrently with Math 654.
Reading Course: I will also offer this as a reading course (Math 685) for students in the distance Master's program.
Please see web page for that option.
I have assigned some homework.
Here are links to snippets of computer code used in class.
Geometry seminar Mondays at 15:00 and Fridays at 16:00 in Bloc 628
Special Mini-Course on Toric varieties by Jose Ignacio Burgos Gil, TBA between 4 and 14 February.
Texas Algebraic Geometry Symposium, February 8–10, UT-Austin.
CombinaTexas, March 23–24, TAMU.
Expected topics to cover:
  • Algebraic-geometric dictionary
  • Resultants and elimination
  • Gröbner bases, including algorithms based on Groebner bases
  • Solving polynomial systems symbolically
  • Solving systems of polynomial equations using numerical continuation
  • Certification of numerical solutions. Smale's α-theory
  • Numerical algebraic geometry. Witness sets and numerical irreducible decomposition
  • Real root counting. Sturm's theorem. Fewnomial theory
  • Toric ideals
  • Toric degenerations and Khovanskii bases
Course description:
    This course will cover the basics of computational algebraic geometry, including the core algorithms in the subject, as well as introduce some of the most common algebraic varieties which occur in applications. We will gain familiarity with software for algebraic geometry, including the systems Macaulay 2, Singular, Bertini, and PHCpack. Students will complete a final project in the subject which will be presented to the class in lieu of a final exam. Grading will be based on final projects and some written/computer work through the term.
Textook:
This will be chapters from a book I am writing on Algebraic geometry for applications with Thorsten Theobald:
Notes on The Algebraic-Geometric Dictionary Revised version of 14 January 2019
Notes on symbolic computation.
Notes on Properties of Varieties.
Notes on Numerical Algebraic Geometry.
The last two are (Dec 2018) a bit incomplete. I plan to complete them for this course.
You are also welcome to get your hands on the the award-winning Ideals, Varieties, and Algorithms by David Cox, John Little, and Donal O'Shea. In particular it contains all the algebraic preliminaries, and is a great resource. It assumes minimal preprequisites, and may not be completely at the level of an advanced graduate class, but it is well worth reading. It is also available in a free download from the TAMU library website!
Last modified: Thu Jan 31 13:17:21 CST 2019