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:
Hilbert's Basis Theorem;
ideal membership problem;
elimination theory and resultants;
the correspondence between ideals and varieties;
irreducibility of varieties;
Hilbert's Nullstellensatz;
projective varieties and homogeneous ideals;
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: I originally learned this material from
D. Cox, J. Little, D. O'Shea,
Ideals, varieties, and algorithms:
An introduction to computational algebraic
geometry and commutative algebra, Second edition,
Undergraduate Texts in Mathematics,
Springer-Verlag, New York, 1997.
This has been ordered by the bookstore and will be on reserve in
the library soon. A more advanced account can be found in
D. Eisenbud,
Commutative algebra:
With a view toward algebraic
geometry, Graduate Texts in Mathematics, 150,
Springer-Verlag, New York, 1995.
Contact information:
Brendan Hassett
Office: Herman Brown 422
Phone: (713) 348-5261
Email: hassett@math.rice.edu
Webpage:
http://www.math.rice.edu/~hassett