MATH 499: VIGRE Computational Algebraic Geometry Junior Seminar
Fall 2007

Singularities of Plane Curves


y^2-x^3+12x-16Instructor: Christian Bruun
Office: Herman Brown 42
Office Hours: TBD
Tel: (713) 348-8304
Email: cbruun@rice.edu
Time: Wed 12 pm – 1 pm
Location: HB 453

Schedule

Project Suggestions

 


Announcements

·         September 5: There will be an informational meeting at 12 pm in HB 453.  All students interested in the seminar are encouraged to attend.

·         October 17: There will not be a seminar meeting due to Midterm Recess.

·         November 19: VIGRE II site visit.

·         December 7: Chris's talk, 12 pm – Elliptic Curves and the Congruent Number Problem (abstract).

·         December 13: Rob's talk, 1 pm – Geometric Theorem Proving (abstract).


Prerequisites:

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 taken MATH 211 Ordinary Differential Equations, MATH 212 Multivariable Calculus or MATH 355 Linear Algebra, or who have some basic knowledge of matrices and experience with mathematical proofs 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

·         Gröbner basis calculations

·         Projective space and homogeneous equations

·         Resolution of singularities

·         Analytic equivalence of singularities

·         Classification of plane curves

 


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.

·         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.

 


References:

·         Robert Bix: Conics and Cubics

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

·         Cox, Little, O'Shea: Using Algebraic Geometry

·         Joe Harris: Algebraic Geometry: A First Course