# MAG

Welcome to **m**etauni **A**lgebraic **G**eometry (MAG). This is a series of online algebraic geometry classes taught at metauni, using Roblox for the 3D virtual environment and voice chat and Discord for community. The first course MAG1 ran from 16/6/2022-11/8/2022.

**When**: starting June 16 2022, weekly 1.5 hour lectures on Thursdays 7:00-8:30am AEST, weekly 1 hour exercise sessions (see below for times). The class will run for 8 weeks.**Where**: The Rising Sea, a 3D virtual world built in Roblox, which is part of metauni. At metauni we write on blackboards (which retain their contents when you leave) and talk using position-based voice chat (people far away can’t hear you). We also use the metauni Discord. See the instructions for how to set it up.**Why**: to introduce the beauty of algebraic geometry in a hands-on way, emphasising the link between classical ideas and modern computation.**What**: the textbook is D. A. Cox, J. Little, D. O’Shea “Ideals, Varieties, and Algorithms” 4th edition, referred to below as**CLO**. You don’t need a copy as we’ll post our own lecture notes based on the book, but it’s a great book and we recommend getting a copy.

The *who* is a community of volunteers:

**Lecturers**: Ken Chan and Dan Murfet, both with PhDs in algebraic geometry.**Tutors**: Edmund Lau, Rohan Hitchcock and Will Troiani (and you?).

**Subscribe to the mailing list** to receive updates on future classes (if you keep an eye on the `#mag`

channel in the metauni Discord, no need to sign up here).

## Course content

There will be a two week pre-course

**Pre-course 1:**Where do lines meet? (solving linear equations by Gaussian elimination)*2-6-22*(video).**Pre-course 2:**Functions and set theory, long division for integers*9-6-22*(video).

Each lecture will be 1.5hr. In the schedule below, `S1.3`

refers to Section 1.3 of CLO, `(K)`

means Ken and `(D)`

means Dan.

**Lecture 1:**What is algebraic geometry?`(D) S1.1, S1.2, S1.3`

*16-6-22*(notes, video).**Lecture 2:**Ideals and polynomial division`(K) S1.4, S1.5`

(video).**Lecture 3:**Monomial orderings`(D) S2.1, S2.2`

(notes, video).**Lecture 4:**Division algorithm`(K) S2.3`

(video).**Lecture 5:**Dickson’s lemma`(D) S2.4`

(notes, video).**Lecture 6:**Hilbert basis theorem`(K) S2.5`

(video).**Lecture 7:**Buchberger’s criterion`(D) S2.6, S2.7`

(notes, video).**Lecture 8:**Buchberger’s algorithm`(K) S2.6, S2.7`

(video).**Lecture 9:**Elimination theory`(K) S3.1, S3.2`

(video).

All videos can be found in the MAG playlist.

## Exercise sessions

Mathematics is not a spectator sport: you learn by doing. Each week there will be assigned exercises, some marked *Foundation* and others marked *Extension*. You are expected to attempt all the Foundation exercises. There are three weekly 1hr exercise sessions:

- Thursday 08:30-09:30 AEST
- Thursday 16:00-17:00 AEST
- Friday 18:00-19:00 AEST
- Sunday 08:00-09:00 AEST

You should join the main MAG venue for these sessions, where you will be working with other students at the virtual boards. Each week at least one of these sessions will be supervised by an experienced volunteer tutor (that session will be advertised on Discord). It is recommended you work on the problems outside of exercise sessions as well.

Problems are divided into **Foundation Example (FE)**, **Foundation Proof (FP)**, **Extension Example (EE)**, **Extension Proof (EP)**. Students are expected to attempt all foundation exercises. References are to **CLO** with `x.y.z`

meaning Chapter `x`

, Section `y`

, Exercise `z`

.

**Week 1**(Will Troiani @ Friday session): FE 1.1.5, FE 1.2.7, FP 1.1.6, FP 1.2.6, EE 1.3.16 (for fans of Bezier), EP 1.2.15**Week 2**(Edmund Lau @ Thursday 8:30): FE 1.4.9, FE 1.4.10, FE 1.4.11, FP 1.4.4, FP 1.4.7, EE 1.4.12, EP 1.4.6.**Week 3**: (Rohan Hitchcock @ Thursday 16:00): FE 1.5.9, FP 1.5.10, EE 1.5.17, EP 1.5.12.**Week 4**: (Will Troiani @ Friday session): FE 2.2.1, FP 2.2.5, FP 2.2.8, FP 2.2.11, EP 2.2.12, EE 2.1.4.**Week 5**: (Edmund Lau @ Thursday 8:30): FE 2.4.3, FE 2.4.4, FP 2.4.2, FP 2.4.6, EP 2.4.8.**Week 6**: (Rohan Hitchcock @ Thursday 16:00) FE 2.5.7, FP 2.5.5, 2.5.9, EE 2.5.17, EP 2.5.12.**Week 7**: FE 2.6.2, FE 2.7.1, FE 2.7.2, FP 2.6.4, FP 2.6.11, EE 2.7.9, EP 2.7.8.**Week 8**: FE 3.1.2, FE 3.1.4, FE 3.1.9, FP 3.1.1, EE 3.1.7.

## Illustrations

In a virtual world we can *use the environment* to communicate ideas, using code; we call these *illustrations*.

## Pre-requisites

High school algebra plus a little bit of linear algebra (a basic familiarity with matrices). More precisely, you should have experience working with polynomials in small numbers of variables and low degree. You should know

- How to find the roots of a quadratic polynomial,
- How to plot
`q(x,y)=0`

where`q`

is a quadratic polynomial, - How to differentiate a polynomial,
- How to multiply matrices and do row operations.

You should also be comfortable with functions and terms like domain, codomain, image and preimage, injective, surjective, bijective, union, intersection and complement. If you want to brush up, here are some modules from Khan Academy that cover the necessary background:

- Introduction to Algebra
- Polynomial expressions, equations and functions
- Matrices up to “Row-echelon form and Gaussian elimination”
- Basic set operations.
- Functions.

For tablet and microphone recommends see the hardware page.

## Registration

Registration is now open. Here is the registration process:

- Follow Steps 1 and 2 of the instructions to setup your Roblox and Discord accounts.
- Open the metauni Discord server and post in the
`#mag-registration`

channel with your Roblox username and how you want people to refer to you. This is mainly for your classmates, so you may wish to include a brief description of your interests and reasons for attending the class, and any links to personal webpages or Twitter accounts you want people to see. No pressure though, your Roblox username is enough (for administration purposes we need to know the connection between Discord accounts and Roblox usernames). - Join the metauni Roblox group. This way we can give you permission to write on the blackboards throughout metauni.