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**Instructor:**
Evan
Bullock

**Office: **Herman Brown 408

**Office Hours:** by appointment

**Tel:** (713) 348-2372

**Email:**

**Time:** Thursdays 12:10

**Location:** HB 423

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

- 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.

- Robert Bix: Conics and Cubics
- Cox, Little, O'Shea: Ideals, Varieties, and Algorithms
- Shafarevich: Basic Algebraic Geometry 1
- Joe Harris: Algebraic Geometry: A First Course

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

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