Math 465: Topics in Algebra
Introduction to computational algebraic geometry

Audience: students in mathematics, computer science, physics, and other mathematically-oriented subjects

Description: Algebraic geometry is the application of polynomial algebra to geometry. It is a natural extension of high school analytic geometry, where common geometric objects are represented as the solutions to equations. For example, the circle centered at the origin with radius one corresponds to the locus

{(x,y) : x2+y2=1}.
Modern algebraic geometry applies the tools of abstract algebra to analyze geometric problems. This course will assume the definitions and elementary properties of rings, ideals, and polynomials. A semester course in modern algebra should be more than sufficient background. A highly motivated student could get by with significantly less.

This course will differ from traditional courses in one crucial respect: we will emphasize computational methods as well as the standard abstract mathematical theorems. In particular, Gröbner bases and the Buchberger algorithm will be used to obtain algorithms for many common operations in polynomial ideal theory. At the same time, we will make sure to give complete geometric explanations for the calculations we perform. Hopefully, this should make the subject accessible to students outside pure mathematics.

Here are some specific topics we will explore:

ideal membership problem;
Hilbert's Basis Theorem;
elimination theory and resultants;
the correspondence between ideals and varieties;
irreducibility of varieties and prime ideals;
Hilbert's Nullstellensatz;
projective varieties and homogeneous ideals;
Grassmann varieties and exterior algebra;
Hilbert functions and polynomials;
Bezout's Theorem.

There may be special projects extending beyond these core topics. Graduate students and others interested in the theory will be encouraged to delve more deeply into the commutative algebra lying at the foundation of the subject.

References: The main textbook will be my lecture notes. As an undergraduate, I learned this material from:

Contact information:
Brendan Hassett
Office: Herman Brown 402
Phone: (713) 348-5261
Email: hassett@math.rice.edu
Webpage: http://www.math.rice.edu/~hassett