At the heart of Algebraic Geometry is the ubiquitous need to solve polynomial equations. We will study the interplay between the discrete algebraic structures on polynomials and the continuous geometrical objects which are sets of solutions to polynomial equations. We will emphasize algorithms and computational techniques over more abstract aspects of the subject. The goal will be to discuss applications to robotics, motion planning, and computer proofs/ automatic theorem proving. Student will be expected to do a focus project either on an application or advanced algorithm implementation. We will follow the beautifully written, award winning textbook by Cox, Little and O'Shea. It starts with the basic concepts of affine varieties, ideals, Groebner bases, Elimination theory, and Zariski topology. We will discuss implementation details for several algorithms including Buchberger's algorithm, ideal membership, resultants, detection of singularities.
Textbook: Ideals, Varieties, and Algorithms; An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox, John Little, and Donal O'Shea. Published by Springer-Verlag in the series Undergraduate Texts in Mathematics, 1991. This book is available electronically from the UW library via a site licence here
Presentations: This topics course will be run like a reading seminar. Students and the instructor will give lectures. A lecture rotation schedule will be established in the first week of the course.
Exercises and Readings from the Text: The single most important thing a student can do to learn mathematics is to work out problems. The second most important thing to do for this class is to read the book. It is a model textbook which every aspiring author should try emulate. Therefore, the goal for the quarter is to read every page and do every exercise in the textbook. Realistically, we will get through as much as we can in 10 weeks. Exercises will not be collected. Instead, each week one team will post solutions and we will have a problem discussion session. Time to be determined.
Projects: Students will be expected to do a focus project either on an application or advanced algorithm implementation. The book offers several suggested topics in Appendix D. This is a chance for us to learn some of the exciting developments since 1991 in this subject. Projects with demos will be presented briefly in the last two lectures of the quarter. A Sage worksheet or similar (Maple, Mathematica, Matlab, ipython etc.) should be submitted on the last day of class including a short summary of the project and a brief demonstration with pictures.
Grading: The grade will be appropriate for a topics course for graduate students. It will be based on the team homework assignment, the team lecture(s), and the individual project.
Computing: Use of computers to verify solutions, produce examples, implement algorithms, and prove theorems is highly valuable in this subject. If you don't already know a mathematical computing platform, then try Sage, Gap, Maple, Mathematica, Macaulay, or Singular.