Simon Fraser University
Engaging the World

Course Outlines

Fall 2020 - MATH 818 G100

Algebra and Geometry (4)

Class Number: 3542

Delivery Method: Remote

Overview

  • Course Times + Location:

    Sep 9 – Dec 8, 2020: Wed, Fri, 2:30–4:20 p.m.
    Burnaby

  • Exam Times + Location:

    Dec 9, 2020
    Wed, 5:00–5:00 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:


This course will be delivered online. You are expected to have access to a reliable internet connection. You will need a computer from which you can download course materials and activities and watch live and/or recorded lectures and participate in live tutorials or workshops.

You will need a camera to take photographs of your work. A phone is acceptable.

Algebraic geometry is a topic central to pure mathematics, and its ideas and results play an ever increasing role in all fields of mathematics. This makes it a very worthwhile topic to learn about if you are interested in mathematics. This course provides an introduction into algebraic geometry at the graduate level with an emphasis on the one-dimensional case: the geometry of algebraic curves.

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 30%
  • Presentation 20%
  • Exam 50%

NOTES:

Lectures will be offered synchronously. A requirement to taking the course is having a good internet connection, capable of handling live (Zoom) video lectures with screen share.

The presentation component of the course will by default consist of a live video presentation by the student, delivered with screen share (pre-made slides recommended).

REQUIREMENTS:

Access to highspeed internet, webcam.

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

RECOMMENDED READING:

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

Reid, Miles. Undergraduate algebraic geometry. London Mathematical Society
Student Texts, 12. Cambridge University Press, Cambridge, 1988. viii+129 pp ISBN: 0-521-35559-1; 0-521-35662-8. Available online through the SFU library: 

https://sfu-primo.hosted.exlibrisgroup.com/permalink/f/1u29dis/TN_cambridge_s10_1017_CBO9781139163699

Hartshorne, Robin. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9. Copy available on reserve at SFU library.

Ravi Vakil. The Rising Sea; Foundations of Algebraic Geometry (December 2015 version). Developing (but very comprehensive) course notes available from http://math216.wordpress.com/

Lenstra's scribe notes, 1995. Available from http://math.berkeley.edu/~bernd/lenstra1995.ps

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 FALL 2020

Teaching at SFU in fall 2020 will be conducted primarily through remote methods. There will be in-person course components in a few exceptional cases where this is fundamental to the educational goals of the course. Such course components will be clearly identified at registration, as will course components that will be “live” (synchronous) vs. at your own pace (asynchronous). Enrollment acknowledges 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. To ensure you can access all course materials, we recommend you have access to a computer with a microphone and camera, and the internet. In some cases your instructor may use Zoom or other means requiring a camera and microphone to invigilate exams. If proctoring software will be used, this will be confirmed in the first week of class.

Students with hidden or visible disabilities who believe they may need class or exam accommodations, including in the current context of remote learning, are encouraged to register with the SFU Centre for Accessible Learning (caladmin@sfu.ca or 778-782-3112).