Math 648: Computational Algebraic Geometry

Instructor: Frank Sottile
Lectures: TΘ 11:10–12:25 Blocker 202
Course webpage: www.math.tamu.edu/~sottile/teaching/15.1/648.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.

Notes for the first week: The Algebraic-Geometric Dictionary
Notes on symbolic computation. These are being revised.

Thursday 26 February: Links to the demonstration files used in class: pentagon.sing quartic.maple quartic.sing sparse.sing.

Homework First homework: Problems from Section of notes.

Tuesday 10 February: Everyone should try three of the problems in Section 2.1, and be prepared to discuss them in class.
Example from the end of Section 2.1.

Projects

April 28 : Kagan Samurkas and Ola Sobieska Free Resolutions and Regular Sequences
April 28 : Kaitlyn Phillipson Smale's 17th problem for binomial and trinomial systems
April 30 : Cameron Farnsworth, Fulvio Gesmudo, and Christian Ikenmeyer Special Concise Tensors
April 30 : Taylor Brysiewicz Numerically Computing Newton Polytopes
6 May 12:00 — 14:00 Emma Owusu Kwaakwah Viro's Construction
6 May 12:00 — 14:00 Yonghui Guan Equations of polynomials of power form
6 May 12:00 — 14:00 Yanhe Huang and Li Ying The Plücker ideal for the Grassmannian
6 May 16:00 — 18:00 Robert Williams and Praise Adeyemo Equations for Schubert varieties
6 May 16:00 — 18:00 Ata Pir and Lanyin Sun Degenerations of toric varieties and regular subdivsions
6 May 16:00 — 18:00 Sean Plummer and Todd Schrader Certified path-tracking of Newton Homotopies
7 May 15:00 — 17:00 Roman Kogan
7 May 15:00 — 17:00 Timo de Wolff Computing Amoebas
7 May 15:00 — 17:00 Eric Wendel Polynomial invariants of a flag of subspaces over a simply-connected nilpotent Lie group

Expected topics to cover:

Algebraic-geometric dictionary
Gröbner bases, including algorithms based on Groebner bases
Resultants and elimination
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 varieties 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, and Bertini. 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.

Last modified: Sat May 2 13:47:01 CDT 2015