MAT 401 - Seminar in Mathematics (Elementary) Algebraic Geometry from an algorithmic point of view

Instructor    Sorin Popescu   (office: Math 4-119, tel. 632-8358, e-mail sorin@math.sunysb.edu)
Location    MAT S-235, TuTh 12:50-2:10

Prerequisites

A good foundation in linear algebra and some knowledge of abstract algebra (such as MAT 313 or MAT 311 or MAT 312). However, I will try to keep the amount of required previous knowledge to a minimum. The seminar will require active student participation and will encourage student discoveries of known (and perhaps unknown) mathematics.

Textbook(s)

Ideals, Varieties, and Algorithms (An Introduction to Computational Algebraic Geometry and Commutative Algebra), by D. Cox, J. Little, D. O'Shea, Springer Undergraduate Texts, Second Edition 1997 This book is a gentle introduction to computational algebraic geometry and commutative algebra at the undergraduate level. It discusses systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms"). The Table of Contents may give you a more detailed picture of the topics covered in the book. Other recommended texts: Undergraduate Algebraic Geometry, Miles Reid, LMS Student Texts 12, Cambridge University Press 1989 Undergraduate Commutative Algebra, Miles Reid, LMS Student Texts 29, Cambridge University Press 1996

I have placed the above books on reserve.

Seminar plan

The seminar will address the following topics:

• Algebra and geometry:
  • Affine varieties (particularly curves in R² and curves and surfaces in R³), parametrizations, ideals.
  • Monomial orderings, division algorithm, Gröbner bases and basic properties. The Buchberger algorithm.
  • Applications of Gröbner bases: elimination theory, singular points, envelope of a family of curves, etc.
  • Some theory on varieties and ideals (an algebra-geometry dictionary)
  • Robotics. Integer programming.
  • Introduction to projective algebraic geometry.
• Use of computer packages:
  Specialized computer algebra systems as well as vizualization tools can help tremendously in the study of certain explicitly given (i.e. by equations) algebraic varieties. We will make use of the following software:
  • Maple and Mathematica for the visualization of implicitly defined curves, surfaces, etc.
  • Macaulay 2 and Macaulay for Gröbner basis calculations, and A local version of the Macaulay 2 manual is available here.

Homework

I will assign problems in each lecture, ranging in difficulty from routine to more challenging. Course grades will be based on these problems and any other class participation; solving at least 2/3 of them will be considered a perfect score. Late homework will be accepted until the second class meeting after the due date, but will not be accepted after that time. Each student will also be required to deliver a 20-30 minute (depending on class size) presentation and hand in one-two papers or computer projects (the first is due by November 1; the second by December 1)

Software documentation, tutorials, and computer projects

We will use the math computer lab in S-235 of the math tower; this lab contains 30 Sun workstations running Unix, as well as a number of PCs running Windows NT. We will be using the Unix machines in class; however, much of the work can be done on other systems. We will rely heavily on Macaulay 2 (a software system devoted to supporting computations in algebraic geometry and commutative algebra) and Maple (a program that can do algebra, calculus, graphics, etc.), although if other tools are better suited to the task, we may make use of them. No previous experience with computers is needed.

Macaulay 2 and Maple are available for most platforms (Unix, Macintosh, Windows, etc); Macaulay 2 can be freely downloaded from the following location, while a student version of Maple can be purchased from Waterloo Maple for $99. You can also use the campus modem pool to dial-in to the mathlab computers.

Here are some important things to read:

• To access the documentation for Macaulay 2 you will need to use a web browser (which you must already be doing if you are reading this). I like the Netscape Navigator, (or its other version, Mozilla, but any other browser such as the Microsoft Internet Explorer is OK if you want. You can get Netscape and MSIE from Instructional Computing, although more recent versions are available from Netscape and Micro$oft.
• We may sometimes have to access documents on the web which are in the Adobe PDF form (Portable Document Format). To read these you will need to have a copy of the freely available Adobe Acrobat Reader on your machine. You can get this from Adobe. For PostScript documents, a free viewer can be downloaded from the GhostScript site.
• Since we will be doing most of our work (in class at least) on a Unix system, you should look over Using Unix. Also useful are the UNIXhelp tutorial, and sections of UNIX is a four-letter word.
• We will use Macaulay 2 in Emacs mode, but we will essentially make no other use of Emacs fancy features. Here is a short Emacs tutorial and here is an Emacs Quick Reference file.
• The simplest way to start Macaulay 2 is to type M2 at the shell prompt. This will run a Macaulay 2 session in the current window. However the recommended way is to run Macaulay 2 as an emacs subprocess, one nice feature of the emacs' Macaulay 2 mode being command completion. Click here for a brief tutorial introduction to the use of emacs with Macaulay 2.

On pages 510 and 512 of the second edition of Ideals, Varieties, and Algorithms, Appendix C mentions computer packages for Maple and Mathematica. These were written mainly for teaching purposes and tend to be rather slow in comparison with a dedicated system as Macaulay 2. You may browse/download these packages from David A. Cox's web site. For Maple V, Release 5(1) you may also download the package, a tutorial and a reference worksheet from the following location:

Package for Release 5.1 of Maple V Tutorial Worksheet for the Package Reference Worksheet for the Package

Maple Package for Maple 6 Tutorial Worksheet for the Package (Maple 6) Reference Worksheet for the Package (Maple 6)

Macaulay 2 is a software system devoted to supporting research in algebraic geometry and commutative algebra, developed by Daniel R. Grayson and Michael E. Stillman. Here are two elementary tutorials:

• A short tutorial introducing a number of basic operations using Groebner bases, and at the same time a range of useful Macaulay 2 constructs.
• A chapter (click here for PDF) by Bernd Sturmfels on elementary computations in Algebraic Geometry from a forthcoming book on Macaulay 2. It illustrates also the use of Macaulay 2 for some of the computations in the textbook by Cox, Little and O'Shea.

Worksheets, handouts, and other class related materials:

Download class related materials from the following links:

• August 30, 2001
• September 18, 2001
• September 25, 2001
• October 2, 2001
• October 9, 2001
• October 11, 2001
• October 16, 2001
• November 1, 2001

Links

Here are some links related to (elementary) uses of Gröbner bases:

• At the Geometry Center at the University of Minnesota, there is a lab on the nephroid (a kidney-shaped curve with some interesting mathematical properties) You may browse the entire lab, starting at the introduction, or read just the part which makes use of Gröbner bases (see also Projections, Profiles, and Envelopes and Singular Sets of Algebraic Curves and Surfaces).
• The Computer Algebra Information Network (based in Europe) has a lot of interesting material on Gröbner bases. For instance take a look at the various tutorials.

Special needs

If you have a physical, psychiatric, medical or learning disability that may impact on your ability to carry out assigned course work, you may contact the Disabled Student Services (DSS) office (Humanities 133, 632-6748/TDD). DSS will review your concerns and determine, with you, what accommodations may be necessary and appropriate. I will take their findings into account in deciding what alterations in course work you require. All information on and documentation of a disability condition should be supplied to me in writing at the earliest possible time AND is strictly confidential. Please act early, since I will not be able to make any retroactive course changes.