How to learn pure mathematics on your own: a complete self-study guide

Author: Aleph 0

This video has a list of books, videos, and exercises that goes through the undergraduate pure mathematics curriculum from start to finish.


Watch this for a flavor of what pure mathematics is like:
(Fredrich Schuller’s Lectures on Differential Geometry and Topology)

I watched these when I was a high-schooler, curious about what pure math was. Even though I understood very little, they fascinated me beyond measure!


Open letter:
Book: “Understanding Analysis” by Stephen Abbott.
Videos: MAT137 Playlist (


Book: “Linear Algebra Done Right” by Sheldon Axler
Problems: “Linear Algebra” by Insel, Freidberg, and Spence
Videos: Sheldon Axler’s Playlist


Book: “Topology through Inquiry” by Su and Starbird
Online Notes with Problems: MAT327 Course Notes (
Videos: Point Set Topology Playlist ( and Algebraic Topology Playlist (


Book: “Differential Equations with Boundary Value Problems” by Zill and Cullen

I recommend focusing on these sections:
Chapter 1 (Introduction!)
Chapter 4.1 (Preliminary Theory of Linear Equations)
Chapter 4.3 (Homogeneous Linear Equations with Constant Coefficients)
Chapter 7 (Laplace Transform)
Chapter 8 (Systems of Linear Differential Equations)
Chapter 9 (Numerical Methods)
Chapters 11, 12, 13 (Fourier Series and Partial Differential Equations)


Books: “Visual Complex Analysis” by Tristan Needham
Videos: Wesleyan University Playlist (

EDIT: In hindsight, I think the best book to learn complex analysis is “A Friendly Approach to Complex Analysis” by Sara Maad and Amol Sasane. It’s really friendly (as the title suggests!) and has tons of practice problems with solutions. I found this book a lot easier to go through than Needham’s book.


Book: “Contemporary Abstract Algebra” by Gallian
Videos: Socratica Abstract Algebra Playlist (

EDIT: For a more in-depth video series, check out these playlists below. They’re by far the most comprehensive video series I’ve found so far.

(Group Theory)
(Ring and Field Theory)


Book for Intuition: “A Geometric Approach to Differential Forms” by David Bachman
Book for Rigor: “Introduction to Manifolds” by Loring Tu
Videos: WhyBMaths (

I’d love to hear if you learn one of these subjects during lockdown. Let me know in the comments below how it goes. Stay safe and happy learning!

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Intro: (0:00)
Real Analysis: (1:38)
Linear Algebra: (2:42)
Topology: (3:27)
Differential Equations: (4:34)
Complex Analysis: (5:22)
Abstract Algebra: (5:51)
Differential Geometry: (6:36) .

Video source Youtube

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