MATH 499: VIGRE Computational Algebraic Geometry Junior Seminar (Spring 2011)

Singularities of Plane Curves


Instructor: Evan Bullock
Office: Herman Brown 408
Office Hours: Wed 3:30-5:00pm, Thu 1:15-2:15pm
Time: Mondays, 12:00-12:50pm
Location: HB 453

Homeworks & Notes


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.


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:


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.


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

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.

Return to Evan Bullock's web site or to Rice's VIGRE PFUG: Computational algebraic geometry.

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