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

1. Real Analysis
2. Linear Algebra
3. Point Set Topology
4. Complex Analysis
5. Group Theory / Ring Theory
6. Galois Theory
7. Differential Geometry
8. Algebraic Topology

These topics are roughly arranged in order of increasing difficulty. It’s fine to skip around a lot (I definitely did when I was learning this stuff) but a lot of these topics have earlier topics as prerequisites. For example, Topology and Group Theory are prerequisites to learning Algebraic Topology.

Some notes:

1. This is 4 years of content. It covers the standard undergraduate math curriculum. So it’s not really something that can 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.

Let’s get into it!

Real Analysis

• Book: Understanding Analysis by Stephen Abbott – A beginner-friendly introduction to rigorous calculus, offering clear explanations and an intuitive approach.
• Videos: Lectures by Francis Su – Engaging and accessible lectures that clarify abstract concepts with real-world examples. Playlist here.

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.