Syllabus

Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics! In one respect this last point is accurate… —David Mumford, 1975

La découverte est le privilege de l’enfant: l’enfant qui n’a pas peur encore de se tromper, d’avoir l’air idiot, de ne pas faire sérieux, de ne pas faire comme tout le monde.
[Discovery is the privilege of the child: the child who has no fear of being wrong again, of looking like an idiot, of not being serious, of not doing things like everyone else.] —Alexander Grothendieck, 1986

Overview

Algebraic geometry could be defined as the branch of mathematics concerned with describing systems of polynomial equations, just as linear algebra is the branch concerned with systems of linear equations. This definition is short, accurate in some respects, and makes an important analogy: one of the key ideas we’ll explore in this class is a kind of polynomial generalization of the row reduction algorithm for systems of linear equations. That being said, this definition also misses an important point: algebraic geometry is a bustling intellectual crossroads! Number theorists, combinatorialists, complex analysts, representation theorists, string theorists, computer scientists, statisticians, and many others have reason to interact with the subject. Our goal in this class is to wade into these vast and deep waters.

Course Materials

Textbook

See here.

Zulip

Zulip is chat software (a bit like Discord, Slack, or Piazza), and it will be our primary means of communication for this class. It’s open-source and you can use it in a browser or by installing an app on any platform. I’ve set up a Zulip organization with a dedicated stream for our class. You can find an invite link on Canvas. Register using any name you’d like to use for this class, but you must use your official ...@ucsd.edu email address as it appears on the class roster. If you don’t have access to the invite link or you don’t have a ...@ucsd.edu email address, please reach out to me to explain the situation.

Use the Zulip class stream to ask any content-related and logistics-related questions you have. Certain assignments for the course will ask you to post things to our class stream regularly. I will also use Zulip to make class-related announcements. If you have private questions about situations specific to you, please use Zulip to send direct messages (instead of emails or Canvas messages, which I may not respond to).

Sage

At a few points in the quarter, you may find that you would prefer not to do certain computations by hand. If you find yourself in this position, I suggest playing around with Sage.

Course Structure

Philosophy

No class (or classes) will ever teach you everything you might need to know about a subject. In the long run, the most important thing to learn is how to learn independently: in pedagogical jargon, that’s “how to be a self-regulated learner.” Research in education suggests that the following three things are key aspects of learning how to be a self-regulated learner of math:

1. Active reading. Reading math can be very different from other kinds of reading. You have to stop constantly to work out exercises and examples yourself, instead of just reading through them. You have to doodle pictures to make sure you have some kind of an image in your head of what’s going on. You have to try to formulate precise questions about things you don’t understand.

2. Reciprocal teaching. Talking to your peers about math is important. If you don’t understand a particular concept, you’re more likely to get an explanation that you actually find helpful from a peer. If you think you do understand a particular concept and help a peer who’s struggling, you’ll almost certainly find that the process of explaining the concept will solidify your own understanding of it. Learning happens best in community, and it is in your best interest to make sure you have a mathematical community.

3. Metacognition. A key part of learning how to learn is reflecting on your learning and taking the time to ask yourself questions about your learning. What parts of your study habits are working well for you? What parts aren’t working and how can you change these parts? What kind of a mindset do you have towards math, and what can you be doing to help cultivate a growth mindset in yourself?

Practicing these three things is built into course structure. It may benefit you to bear these three things in mind as you go through the course.

Class

You’ll be expected to read and make a preliminary attempt to understand material on your own before coming to class. Attendance is strongly encouraged, but it is not required. The class will not be podcasted.

Assignments

There will be a few different types of assignments in the course:

• Reading Assignments: These involve reading something, attempting some exercises, and formulating a question about the content. One of these will be due before most class sessions.

• Reflection Assignments: These ask you to reflect on various aspects of your relationship with mathematics, your mathematical learning, etc. You’ll have about one these due every week.

• Project: You’ll do a project. It’ll be due at the end of the quarter.

Assignment deadlines are indicated on the schedule. Late assignments will not be accepted.¹

Quizzes

There will be 5 quizzes over the course of the quarter. There will be no make-up quizzes.

Grades

It’s unfortunate that grades exist, but they do. Here’s how they’ll be determined in this class.

First, numerical scores will be calculated as follows:

Your numerical score will be then converted to a letter grade using the following cutoffs:

At the end of the quarter, I may decide to lower some of the above cutoffs, but I will not increase them.

Accommodations

If you experience disability-related barriers to your learning, please contact the Office for Students with Disabilities right away to have them provide a current Authorization for Accommodation (AFA). The AFA should be received at least one week in advance of the requested accommodations, and ideally by the end of the first week of classes. We may be unable to accommodate late requests.

Integrity

Act with integrity. You’ll learn more, and you’ll be practicing good habits for ethical decisions that you’ll have to make for the rest of your life. If you have a question about whether something class-related is integrous, just ask first. Academic integrity violations have to be reported and result in serious consequences, none of which is any fun for anyone involved — so please, just don’t do it.

Diversity

The pursuit of knowledge thrives in diverse and inclusive environments. I expect that all of us, myself included, will work towards making our class a welcoming space for everyone, no matter how we might identify on race, ethnicity, nationality, socioeconomic background, gender, sexual orientation, ability, age, and the many other dimensions of identity. I also encourage you to reach out if there are ways to have our space be more comfortable for you. If anyone says anything in class that makes you feel unwelcome, please let me know.