# Math 499 Project

Each student is required to give a twenty-five minute presentation on some topic related to the seminar. You must also provide an abstract for your talk at least a week in advance. The aim of this project is to examine a particular topic related to plane curves or algebraic geometry in more depth.

Here are a few project ideas:

• Symmetric polynomials

• Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms;

• Elliptic curves (characterize them, algebraic structure, analytic properties,etc.)

• Source: Koblitz: Introduction to Elliptic Curves and Modular Forms

• Blowing-up (plane curve) singularities

• Source: Harris: A First Course in Algebraic Geometry

• Newton polygons and Puiseux expansions

• Source: Brieskorn, Knorrer: Plane Algebraic Curves

• Genus of a curve, or other topological properties

• Source: Miranda, Algebraic Curves and Riemann Surfaces

• Kinematics problem for robotics

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

• Automatic geometric theorem proving

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

• Improvements to Buchberger's algorithm, other algorithmic methods

• Source: Cox, Little, O'Shea: Ideals, Varieties, and Algorithms;
Greuel, Pfister, Bachmann: A Singular Introduction to Commutative Algebra

• Investigate a distinguished curve

These are just a few ideas, with some introductory sources. Your talk should be somehow related to algebraic geometry (and hopefully plane curves), but beyond this, you can research almost anything. You can also see what past seminars have done, along with abstracts, from 2006 and 2005. These should give you some idea of what kind of projects people have done in the past. You might want to study one of these topics, also. Please talk with me about the plan for your project to get some ideas or sources, or if you want some other project ideas.

Back to Math 499.