Modern Algebra

Detailed Information

Course Description

We will study algebraic geometry, one of the oldest and richest areas of mathematics. During the 20th century, the theoretical and very abstract side of the subject was prominent, but with the availability of computers, the computational roots have been reinvigorated. This course will develop the theory behind the computational tools.

What is algebraic geometry? Think back to high-school algebra where you graphed polynomial equations and perhaps found the intersection of plane curves defined by a line and a parabola or more general curves defined by polynomials. Now think about higher dimensional space and consider intersections of hyper-surfaces defined by polynomial equations. Such objects are called algebraic sets or algebraic varieties. What is the dimension? How many components are there? What is the simplest way to describe the intersection? These are some of the geometric questions arising in algebraic geometry.

The fundamental result in algebraic geometry is the algebra-geometry "dictionary" which gives a precise relationship between geometrical objects and algebraic ones: between varieties in n-dimensional space and radical ideals in the polynomial ring in n variables. Algebra provides tools for formalizing and being precise about geometric concepts, which can be rather intuitive. Conversely, algebraic results have a geometric interpretation that brings richness to abstract formulas.

The fundamental tools in computational algebraic geometry are Grobner bases for ideals and Buchberger's algorithm. Grobner bases are a generalization of the greatest common divisor of integers. Just as the Euclidean algorithm may be used to compute the gcd, Buchberger's algorithm is used to compute a Grobner basis for an ideal.

In the last few decades, numerous applications of algebraic geometry have been discovered: in coding theory, cryptography, robotics, object recognition, engineering, genomics etc. Some links that show the scope of recent work are: The Society for Industrial and Applied Mathematics Activity Group on Algebraic Geometry, The Special Semester on Grobner Bases and Related Methods; The Thematic Year on Applications of Algebraic Geometry at the Institute for Mathematics and Its Applications; and the work of Bernd Sturmfels. Powerful computational software has also been developed. See for example Sage , Macaulay 2, Singular, and Magma. These computational tools are of great importance in application.

Required Materials

Cox, Little, O'Shea Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra 2nd Ed., Springer-Verlag, 1997, or 3rd edition 2007.

William A. Stein et al. Sage Mathematics Software The Sage Development Team, 2011, http:www.sagemath.org

SDSU Sage Tutorial

The text is a well written book that is one of the standard references in computational algebraic geometry. I will cover the core material on Grobner bases (chapters 1-3), the algebra-geometry dictionary (chapter 4) and, in less detail, functions on a variety (chapter 7), and projective space (chapter 8). Student interest will also guide the course.

Sage is an open source mathematics software package that incorporates numerous other open-source packages into a unified package. The Sage tutorial will help you get started.

Prerequisites

• Properties of the Integers: The division theorem and divisibility, the Euclidean algorithm, unique factorization, modular arithmetic.
• Polynomial Ring in One Variable: The division theorem, greatest common divisor, the Euclidean algorithm, unique factorization. The correspondence between factors and roots. Polynomial rings modulo a polynomial.
• Commutative Rings and ideals: The general language of rings and ideals. Integral domains, the quotient of an integral domain by an ideal, homomorphisms.
• Linear Algebra: nullspace, subspace, dimension, basis.

Grading

There will be one exam.

There will be a final project, with a great deal of latitude in choice of topic. You may focus on theoretical questions, implementation of an algorithm, an applied problem, or some combination. I have plenty of references, including recent research, that should be accessible to you by the end of the course. You may also develop an educational module for advanced high-school students. More information about the project will be provided later in the semester. The final grade will be apportioned as indicated in the table +/- 5 points for each item.

Problem Sets	40%
Midterm	25%
Final Project	35%