Winter 2007
Math 689, Section 4: Introduction to Applicable Algebraic Geometry

Instructors: Luis David Garcia-Puente and Frank Sottile.

This graduate course is being co-taught by Luis Garcia and Frank Sottile. Frank will lecture the first six weeks (until 23 February) and Luis will lecture the remainder of the term, with some guest lectures by Sottile and others later in the term. This page is Sottile's Home page for the course.
Course Announcement.

Homework is due 28 February. Get it here.
Frank Sottile
  Office: Milner 303
Telephone: 845-4169
Email: sottile"at"math.tamu.edu
WWW: www.math.tamu.edu/~sottile
Weekly schedule
Frank has no scheduled office hours. Schedule an appointment with him or drop by his office when his door is open.
  Some notes from the course.
    Frank does welcome suggestions and comments on these notes.
Some Older notes of Frank's.
Lectures: MWF 9:10 -- 10:00 AM in Blocker 155.
Recommended Texts: Some of the matrial in the course is found in (parts of) the following four books.
  • Cox, Little, and O'Shea, Ideals Varieties and Algorithms,
    Special Note about this book: Springer is putting out a new edition in February. However, they have a special emergency printing of the previoyus edition (which is just fine). If you cannot get immediate delivery from Amazon, call Springer, mention this fact (the emergency printing). If necessary, say that "Ann Kostant told Professor Sottile about theis special emergency printing of this book". I can help if necessary.
  • Cox, Little, and O'Shea, Using Algebriac Geometry, This has a special paper back edition!.
  • Sturmfels, Gröbner Bases and Convex Polytopes,
  • Sturmfels, Solving Polynomial Equations,
Course webpage: www.math.tamu.edu/~sottile/teaching/07.1/689.html
Course Description
Algebraic Geometry, the geometric study of polynomial equations, has a long history as a core discipline within pure mathematics. In recent years it has been finding many new applications outside of mathematics. For example, the IMA (Institute of Mathematics and its Applications) is devoting the current academic year to a program on applications of algebraic geometry, and Texas A&M will be hosting a summer school during July and August on Applicable Algebraic Geometry, with 30-50 students attending from across the United States. Several TAMU faculty work in this growing area.

The purpose of this Spring 2007 course will be to give students an introduction to the applicable side of algebraic geometry. We will see the basic concepts of commutative algebra and algebraic geometry that will be assumed prerequisites for the summer school. This course may also be used as preparation for the more comprehensive algebraic geometry course that may be offered next year. Along the way, we will show how some tools from commutative algebra and algebraic geometry can be used to solve systems of polynomial equations, you will see many fundamental classes of varieties that arise in algebraic geometry, and you will be exposed to some of these new applications of algebraic geometry.

Prerequisites
Linear algebra and abstract algebra. Graduate students from Engineering and Computer Science are welcome to register; We can fill in any gaps in your knowledge.
Topics
Polynomials, ideals, varieties
  1. Algebra-geometry dictionary
  2. Varieties
  3. Gröbner bases
Systems of polynomial equations
  1. Elimination and resultants
  2. Gröbner basis conversion
  3. Finding real roots
  4. Bounds on number of solutions
Ideals and Varieties
  1. Monomial ideals
  2. Toric Ideals
  3. Toric varieties and polytopes
  4. Determinantal Ideals
Applications
  1. Optimization
  2. Integer programming
  3. Statistical inference
  4. Geometric Modelling
Grading
There will be two or three problem sets collected and graded during the term, and all students will make a presentation in the last few weeks on some additional topic. We will discuss this later in the course.
Modified since 2 January 2007