Math 480, Computational Algebraic Geometry

Math 480 -- Computational Algebraic Geometry

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Course Textbook

The course mainly follows Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, however other textbooks can also provide supplementary material.

• Algebraic Geometry notes by Ravi Vakil
• Introduction to Algebraic Geometry notes by Igor V. Dolgachev

Overview

One may read about the history of Algebraic Geometry in the Wikipedia article on the subject. There is also thorough examination of the history of the subject written by Dieudonne, which is available here.

Algebraic Geometry is a very active research field -- a quick glance at the recent additions to the arXiv supports this assertion. There are quite a few research opportunities for undergraduates interested in Algebraic Geometry, especially in the North-Eastern region of the US. AGNES is a series of biannual weekend workshops in algebraic geometry. MIT and Harvard host a Algebraic Geometry seminar. Oakland University in Michigan organizes a yearly conference in Algebraic Geometry (MCAG). SIAM AG hosts a Algebraic Geometry mailing list. Finally, there is a REU program in Algebraic Geometry at Clemson University in South Carolina.

Course Structure

A typical semester spent on this course will reach Chapter 7 of the course textbook, "Invariant Theory of Finite Groups". Progress through the material can either be measured weekly through the completion of programming assignments or short presentations of material. A source code repository for the course is available here.

• Chapter 1: Geometry, Algebra, and Algorithms
  • Programming Assignment:

Write a python program which can determine whether a polynomial is within the ideal generated by a given list of polynomials.

• Chapter 2: Groebner Bases
  • Programming Assignment:

Implement the multi-variable division algorithm in Python.

Implement Buchberger's Algorithm.

• Chapter 3: Elimination Theory
  • Programming Assignment:
• Chapter 4: The Algebra-Geometry Dictionary
  • Programming Assignment:
• Chapter 5: Polynomial and Rational Functions on a Variety
  • Programming Assignment:
• Chapter 6: Robotics and Automatic Geometric Theorem Proving
  • Programming Assignment:
• Chapter 7: Invariant Theory of Finite Groups
  • Programming Assignment: