Simon Fraser University
Engaging the World

Course Outlines

Fall 2022 - MATH 818 G100

Algebra and Geometry (4)

Class Number: 6338

Delivery Method: In Person

Overview

  • Course Times + Location:

    Sep 7 – Dec 6, 2022: Wed, Fri, 2:30–4:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

An introduction to algebraic geometry with supporting commutative algebra. Possible topics include Hilbert basis theorem, Hilbert's Nullstellensatz, Groebner bases, ideal decomposition, local rings, dimension, tangent and cotangent spaces, differentials, varieties, morphisms, rational maps, non-singularity, intersections in projective space, cohomology theory, curves, surfaces, homological algebra.

COURSE DETAILS:

Algebraic geometry is the study of polynomials in more than one variable. This very old field of study has many connections to other areas of mathematics, and the modern perspective has a commutative algebra framework. To paraphrase Sophie Germain: "Algebra is nothing but written geometry; geometry is nothing but drawn algebra".

This is an introduction to algebraic geometry at the graduate level with an emphasis on the geometry of algebraic curves, which will prepare students for the Spring 2023 offering of Math 819.

Topics will likely include:

  •     basic notions of affine and projective varieties
  •     the notions of rational maps between varieties
  •     differentials and singularities
  •     function fields
  •     divisors, the Riemann--Roch theorem
  •     classification of algebraic curves

Grading

  • Assignments 50%
  • Exam 50%

REQUIREMENTS:

This course is delivered in person, on campus. Should public health guidelines recommend limits on in person gatherings, this course may include virtual meetings. As such, all students are recommended to have access to strong and reliable internet, the ability to scan documents (a phone app is acceptable) and access to a webcam and microphone (embedded in a computer is sufficient). 

Materials

REQUIRED READING:

Shafarevich, I. R. Basic algebraic geometry, part 1: Varieties in Projective Space; Springer-Verlag. Available online through the SFU library: 

https://sfu-primo.hosted.exlibrisgroup.com/permalink/f/usv8m3/01SFUL_ALMA51189504030003611

William Fulton, Algebraic Curves, freely available, see: Fulton's homepage.

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

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 website 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