My Ultimate Self-Study Guide to Learning Pure Math - Part 1

+ with part 2 coming soon!

A while back, I made a YouTube video explaining how to self-study pure math:

To date, it’s the most viewed video on the channel.

Since then, I’ve received a lot of emails asking for more detailed learning resources. So in the next few issues, I wanted to collate this information and share it with you.

Here are the topics I’ll be covering in this guide:

1. Linear Algebra

2. Point Set Topology

3. Group Theory

4. Differential Geometry

Some general tips

But before we dive in, I wanted to share some general tips that helped me when learning math.

1. Learning math takes time. The subjects I’ll share today cannot really be studied in a few days’ time. Each subject would take 4 months to really study well.

2. Do as many problems as possible. So for each topic, I’ve included resources (most of which are freely / cheaply available) that include a lot of practice problems.

3. Re-read these books many times. I normally only really understand a textbook after having read it 4 times. Most people are the same.

4. If possible, find a friend to explain things to. I learn better by explaining things to people. I expect most people are the same!

Let’s get into it!

Linear Algebra

• Book: Linear Algebra Done Right by Sheldon Axler – Focuses on vector spaces and linear transformations, offering a fresh perspective without relying heavily on determinants.

• Videos: Sheldon Axler’s Playlist – A concise lecture series that complements the book’s content. Playlist here.

Point Set Topology

• Online Notes: MAT327 Course Notes – Comprehensive notes with problems, ideal for building a strong foundation in topology. Available here.

Group Theory

• Book: Topics in Algebra by I.N. Herstein (Chapter 2) – An accessible yet rigorous introduction to the foundations of group theory.

• Videos: Lectures by Benedict Gross – Clear and engaging lectures that explore group theory concepts step by step. One of my favorite lecture series!

Differential Geometry

• Book: Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby. A thorough introduction to the concepts and tools of differential geometry, suitable for advanced learners.

• Videos: Online Lectures by Fredrich Schuller. These were the lecture videos that first got me interested in math!

Challenge Problem

Here’s this week’s challenge problem:

The integers 1, ..., n are arranged in any order. In one step you may switch any two neighboring integers. Prove that you can never reach the initial order after an odd number of steps.

Got a solution to the challenge problem? Submit it here.

Solution from last week

See here for the solution to last week’s problem.

Huge thanks to Charles Burns (Oxford) and Chris (Italy) for submitting solutions to last week’s challenge problem.

Thanks for reading and happy learning! See you next week,

Adithya