# 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!

REAL ANALYSIS

Open letter: http://assets.press.princeton.edu/chapters/s10825.pdf
Book: “Understanding Analysis” by Stephen Abbott.

LINEAR ALGEBRA

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

TOPOLOGY

Book: “Topology through Inquiry” by Su and Starbird
Online Notes with Problems: MAT327 Course Notes (http://www.math.toronto.edu/ivan/mat327/?resources)

DIFFERENTIAL EQUATIONS

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)

COMPLEX ANALYSIS

Books: “Visual Complex Analysis” by Tristan Needham

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.

ABSTRACT ALGEBRA

Book: “Contemporary Abstract Algebra” by Gallian
Videos: Socratica Abstract Algebra Playlist (https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6)

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.

DIFFERENTIAL GEOMETRY

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

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!