Algebraic geometry is the study of polynomial equations and their solution sets. For example, the equation x^2 + y^2 - 1 = 0 determines a circle in the (x,y) plane. When more variables and equations are involved, the solution set may be a complicated set consisting of points, curves, surfaces, and objects of higher dimension. Any set which can be defined by polynomial equations is called an algebraic variety.
Given a variety, one can consider all of the polynomials which are zero on the set. This collection of polynomials is called the ideal of the variety. The ideal is a purely algebraic object while the variety is a geometric one. The interplay between the algebra and the geometry is what makes the subject so interesting. Computer algorithms for analyzing ideals have been developed in the last few decades. Using these, one can study ideals and varieties far too complicated to be worked out "by hand".
The course will introduce the basic ideas of algebraic geometry as well as the computational algorithms. In fact the algorithms can be used to develop some of the theory. In addition, several applications of these ideas will be described.
Ideals, Varieties, and Algorithms (2nd edition), by Cox, Little and O'Shea. This is a very nicely written book which unfortunately contains too much material for one semester. The plan is to cover chapters 1--3 and parts of chapters 4 and 8. Some examples and applications not in the book will also be presented.
|Midterm Exams||1/4 each|
|Midterm I||Friday, October 18|
|Midterm II||Friday, December 6|
|Final Project Due||Monday, December 16|
We will meet once in a mathematics department computer lab for some hands-on experience with the computer algebra system Mathematica. This program is also available on most of the computers at the university. After the initial meeting, you will use the computer on your own for some of the homework problems and possibly for your final project. Lab Handout
This is your chance to explore a part of the subject not covered in class or to put the computational algorithms to work on a problem of your own. The form and topic are very flexible and creativity is encouraged. For some concrete suggestions for projects follow this link: Possible Projects. You could do one of those or, even better, come up with an idea of your own.