This course is geared towards undergraduates who have a strong interest
in mathematics and who wish to get a taste of mathematical research.
Students who have some experience with mathematical proofs and have
taken MATH
212/221 Multivariable Calculus and MATH 355/354 Linear Algebra, or
otherwise have some
basic knowledge of matrices (e.g. MATH 211), should be suitably
prepared for this course.

Description

Modern algebraic geometry is one of the most dynamic and exciting
areas of mathematical research. It manages to incorporate elements from
algebra, geometry, and analysis, as well as a bit of topology, and is
an active area of research in both pure and applied mathematics.
However, even a modestly thorough introduction to the subject requires
a fairly large amount of mathematical machinery. On the other hand, the
problems of classical algebraic geometry are relatively understandable,
and many are solvable with only a moderate knowledge of mathematics. At
the same time, advances in computing power and algorithmic methods in
algebraic geometry have made it possible to quickly and easily analyze
geometric objects in ways that classical geometers would not have
dreamt of. Therefore, while there is still quite a learning curve to
the study of algebraic geometry, students can discover a significant
amount about the subject by attacking some classical problems with a
computational approach.

This seminar will tackle some basic problems in algebraic geometry
using computational methods, but without a heavy emphasis on theory.
The goal is to introduce students to some of the problems and methods
of algebraic geometry without necessarily requiring them to understand
some of the more technical details involved. In particular, we will be
looking at plane curves and plane curve singularities, and performing
calculations to classify various types of plane curves and their
singularities. This will give students an introduction to some of the
basic techniques of algebraic geometry, should they wish to continue in
this area. Also, it will show how computational and algorithmic methods
can be applied to mathematical research.

Each class will consist of a problem solving and presentation
session and a short introduction to new material and techniques. The
first half of each class will be spent on discussing and presenting
problem solutions. During the last half of class, we will introduce new
concepts and techniques, and discuss new problems to work on.

Topics to be covered include:

Polynomials and plane curves

Identifying plane curve singularities

Tangent cone of a curve

Rational parameterizations

Gröbner basis calculations

Elimination theory

Resultants and Discriminants

Grading

This course will be based in large part on student participation. The
majority of each class will be spent on student presentation of the
exercises. In addition, students will be responsible for a short
presentation about a topic of their choice. The last couple weeks of
class will be devoted to these presentations. More credit hours
require a more extensive project including a written component; come
see me if you are interested in signing up for more than one credit.

Attendance/Participation (70%): This course will follow a seminar
format, so attendance is critical. There will be weekly homework
problems to be presented in class, and students will be expected to be
involved with the class discussion.

Project (30%): Each student will prepare
a project and a
twenty-five minute presentation. Topic suggestions will be given during
the first few weeks of class, but projects will most likely consist of
the presentation of a current paper or a chapter or two of an advanced
textbook on some topic relevant to the course. See this list of project suggestions and past projects.

References

There is no textbook for the class, but the following books may be
useful (I would especially recommend Cox, Little, and O'Shea):

Cox, Little, O'Shea: Ideals, Varieties, and Algorithms

Robert Bix: Conics and Cubics

Shafarevich: Basic Algebraic Geometry 1

Joe Harris: Algebraic Geometry: A First Course

Disability Support

If you have a documented disability that will impact your work in
this class, please contact me to discuss your needs. Additionally, you
will
need to register with the Disability
Support Services Office in the
Allen
Center.