Simon Fraser University
Engaging the World

Course Outlines

Spring 2016 - MATH 741 G100

Commutative Algebra and Algebraic Geometry (3)

Class Number: 6419

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 5 – Apr 11, 2016: Tue, Thu, 2:30–4:20 p.m.
    Burnaby

  • Exam Times + Location:

    Apr 21, 2016
    Thu, 8:29–8:29 a.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

A study of ideals and varieties. Topics include affine varieties, ideals, the Hilbert basis theorem, resultants and elimination, Hilbert's Nullstellensatz, irreducible varieties and prime ideals, decomposition of varieties, polynomial mappings, quotient rings, projective space and projective varieties. Additional topics depending on the instructor. Groebner bases and automatic theorem proving in geometry, Bezout's theorem, dimension, and elliptic curves.

COURSE DETAILS:

An introduction to the objects of commutative algebra and algebraic geometry: polynomial rings, varieties (solutions of systems of polynomial equations), ideals, Groebner bases, and quotient rings. This is a generalization of the theory of linear systems and linear algebra to treat systems of non-linear polynomial equations. The discovery of Groebner bases by Bushberger in 1965, followed by the development of software implementations for computing Groebner bases has made possible a very constructive approach to this subject. Throughout the course we will be using Maple for doing calculations.

Ideals and Varieties: Polynomials, ideals and varieties Curves and surfaces in two and three dimensions Parametrizations of affine varieties

Groebner Bases: The division algorithm, the Hilbert basis theorem and Groebner bases Buchbergers algorithm

Elimination Theory Elimination theory, implicitization of curves and surfaces, unique factorization, and polynomial resultants.

Hilberts Nullstellensatz and ideal decomposition Hilberts NullstellensatzIrreducible varieties, prime ideals, maximal ideals Decomposition of varieties and the prime decomposition of ideals Quotient rings

Applications Geometric theorem proving, circle packing problems

Grading

  • Homework 60%
  • Final exam 40%

Materials

REQUIRED READING:

Ideals, Varieties & Algorithms
3/E
Cox, Little & O Shea
Springer-Verlag


ISBN: 9780387356501

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

SFU’s Academic Integrity web site http://students.sfu.ca/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating.  Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the University community.  Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS