Simon Fraser University
Engaging the World

Course Outlines

Spring 2022 - MATH 741 G100

Commutative Algebra and Algebraic Geometry (3)

Class Number: 5872

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 10 – Apr 11, 2022: Mon, Wed, Fri, 4:30–5:30 p.m.
    Burnaby

  • Exam Times + Location:

    Apr 23, 2022
    Sat, 7:00–10:00 p.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:

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. It leads to beautiful interplay between algebra and geometry.

Topics covered will include: polynomials, ideals and varieties, term orders, the division algorithm, Groebner bases and the Hilbert basis theorem, elimination theory, Hilbert's Nullstellensatz, irreducible varieties, projective space, Bezout's theorem.

Grading

  • Assignments (5-10 questions each week; two lowest scores eliminated) 15%
  • Midterm 25%
  • Final Exam 60%

NOTES:

This is a cross-listed course with MATH 441. Students enrolled in the graduate level MATH 741 of this cross-listed course will be assigned additional homework problems in addtion to receiving several extra problems on both the midterm and final exams.

Materials

MATERIALS + SUPPLIES:

Both textbooks are avaiable online through the SFU library.

REQUIRED READING:

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
David A. Cox, John Little, Donal O’Shea
Springer, 4th Edition.

Also available as an e-book from Springer, which can be found here:
https://link.springer.com/book/10.1007%2F978-3-319-16721-3
ISBN: 9783319374277

Introduction to Algebraic Geometry
Brendan Hassett
Cambridge University Press
2007.
ISBN: 9780521691413

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:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity web site http://www.sfu.ca/students/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

TEACHING AT SFU IN SPRING 2022

Teaching at SFU in spring 2022 will involve primarily in-person instruction, with safety plans in place.  Some courses will still be offered through remote methods, and if so, this will be clearly identified in the schedule of classes.  You will also know at enrollment whether remote course components will be “live” (synchronous) or at your own pace (asynchronous).

Enrolling in a course acknowledges that you are able to attend in whatever format is required.  You should not enroll in a course that is in-person if you are not able to return to campus, and should be aware that remote study may entail different modes of learning, interaction with your instructor, and ways of getting feedback on your work than may be the case for in-person classes.

Students with hidden or visible disabilities who may need class or exam accommodations, including in the context of remote learning, are advised to register with the SFU Centre for Accessible Learning (caladmin@sfu.ca or 778-782-3112) as early as possible in order to prepare for the spring 2022 term.