Modern Algebra II

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 polynomials. Now think about higher dimensional space and look at 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 fundamental questions of algebraic geometry. Other classical problems in algebraic geometry are

• classification of curves and surfaces,
• analysis of singularities (points where the variety is not smooth),
• translating from an explicit description of a variety (e.g. a parametrization) to an implicit description (as the zero set of some polynomials) and vice-versa.
• analysis of families of curves.

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. The fundamental notion in computational algebraic geometry is Grobner bases for ideals and the fundamental computational tool is Buchberger's algorithm. Grobner bases are, in a certain way, a generalization of the greatest common divisor of several 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 to show the scope of recent work are: 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 tools have also been developed. See for example Macaulay 2, Singular, and Magma. These computational tools are of great importance in application.

The textbook we are using is very well written and 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. There will be homework assignments with proofs and computional exercises. I may incorporate computer assignments as well. 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.

Required Text

Prerequisites

• Polynomial ring in one variable: The division theorem, greatest common divisor, Euclidean algorithm, the correspondence between factors and roots, unique factorization.
• Integral Domains: ideals, the quotient of an integral domain by an ideal, homomorphisms. The field of quotients of an ideal (I will cover this).
• Linear Algebra: nullspace, subspace, dimension, basis.