# Seminar: Commutative Algebra and Algebraic Geometry

Math 627B
Spring 2017
Course number: 22156
Meeting Tu Th 11:00 - 12:15
GMCS 405
San Diego State University

Professor: Mike O'Sullivan
Web page: http://www.rohan.sdsu.edu/~mosulliv/Teaching/alg-geo17s.html
Email: mosullivan@mail.sdsu.edu
Office: GMCS #579, ext. 594-6697. Also GMCS 413.
Office Hours: Tu Th 1-3:00.
I will usually be available Tu Th 1-2:30. On other days I am often available in the afternoon, but it is best to email me in advance to schedule a meeting. Last Class: Th 5/4.
Final: Tu 5/9.

## Detailed Information

 SCHEDULE ASSIGNMENTS PROJECTS(pdf)

## 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 Groebner bases for ideals and Buchberger's algorithm to compute them. Groebner 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 4th Ed. 2015.

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

The text is a well written book that is one of the standard references in computational algebraic geometry. The authors just won the 2016 American Mathematical Society Steele Prize for Mathematical Exposition We 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). I will include some material on general (commutative) ring theory. 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

A good understanding of the basics of groups, rings and fields (Math 627A is great, Math 521A with some extra algebra expericence is sufficient). Of primary importance are the following topics, which I recommend you review.
• 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.
Take a look at my webpage for Math 521 A. The book by Hungerford is a great reference. Review Chapters 1,2,4,5,3,6 in that order.

## Format

Class time will mix lecture with problem solving. I will be using Matt Anderson's Learning Glass to create short (~15min.) online lectures on some topics. This will free class time for discussion of problems. Be prepared to present your work in class, and also to work on problems in class.