How to self study pure math - a step-by-step guide

As a theoretical physics graduate student, I can strongly recommend the textbook "Differential Geometry, Gauge theories, and Gravity" by Gockeler and Schucker to understand differential geometry with just a physics undergraduate background, where the book introduces but also applies this subject to numerous areas of modern physics - I really recommend it! Also, if you're a physics student and don't really know where to start, I would strongly recommend "Modern Mathematical Physics" By Szekeres which basically introduces from the ground up and assuming no prerequisite all the parts of maths you need to do physics at an advanced level. Hope it helps :)

Should keep me busy for the next 5 years

This video is left as an exercise for the reader.

Great video! As someone who got an undergraduate degree in math, many of the resources here are excellent resources that I made good use of. I look forward to checking out the others to fill in the gaps on some of my weaker subjects.

I would have liked to see this video like 25 years ago. All this material free online. What a wonderful time is today. I want to thank you in the name of humanity.

This makes me very happy. I studied comp sci at uni but have been trying to learn some higher math on my own lately. The specific resource recommendations are extremely valuable to me. I consider this a Christmas gift!

Calculus by Michael Spivak (Third or Fourth edition) is unexpectedly a really good introduction to not just real analysis, but also some number theory and a little abstract algebra at the end of the textbook. It doesn't require any prerequisites, as stated by the author (I think). It also justifies most theorems that are brought up, has proofs for them, and most importantly, an ungodly amount of examples (like 60 per chapter, making up most of the book).

As someone who's doing a degree in mechanical engineering but has a very strong passion for pure math I very much appreciate this video, as well as yours and everyone else's free and amazing videos on many different math topics all across youtube. Keep up the great quality!

How to study math: 1: Read some stuff. 2: Do a lot of problems. 3: Do some more problems. Return to (1).

Thanks for watching! If you have any resources you'd like to recommend, feel free to comment them down below. Also, I have a math newsletter where I collate resources to learn topics in math / machine learning and deliver them to your inbox. If you'd like to sign up for the newsletter, fill out this form: /Rt1f5StAj3yZtakE6

I appreciate how you didn't overload us with resources but gave it straight forward. Much much more appreciated than the channels that throw tons of books or resources.

Hey. I just want to say you've really inspired me to get back into self studying mathematics. I had done a lot of self study in the past, when I had much less mathematical maturity, and got discouraged, in part because I kept skipping prerequisites and the like. Thank you for this and other videos.

I love you. Self teaching math is stupid hard for no reason.

A minor nitpick: the study of differentiable manifolds without a distinguished 2-form (such as a metric or symplectic form) is generally referred to as differential topology, while the inclusion of such a form "elevates" the subject to geometry. So, the point where you say differential geometry ends is actually where differential geometry begins.

Thank you for the video. I wish YouTube had a "love" button. Please don't remove the previous version of this video from the channel. It was so informative and helpful. Just add "2020" or something in the title. I love that video too! Like this comment if you want that video back.

Awesome content. Please cover self study for other fields of mathematics eventually. Thanks for this.

At a starting point i think everyone should read 'Mathematical proofs, a transition into advanced mathematics', and then proceed to read everything you said.

So I noticed the topics for the undergraduate curriculum in this newer version of the video changed slightly but either way it makes me want to bring up a somewhat list of things you would see as an undergraduate in university. First off, not everyone does calculus and differential equations in high school (generally speaking most don't): so if you're going to university, expect those. The topics of Linear Algebra, Real Analysis, and Abstract Algebra are generally the common core of every undergraduate math degree; however, most people will do more things like complex analysis and topology. A lot of American universities will have a class dedicated to learning how to write proofs and learning basic set theory and number theory. Group Theory and Galois Theory aren't usually taught on their own as courses but are rather a large chunk of a first and second semester abstract algebra course respectively. Not everyone does differential geometry or algebraic topology when they're an undergrad but a lot do. Other undergraduate courses include: Probability, Number Theory, Euclidean Geometry, Partial Differential Equations, and basically anything that says "intro to [branch of math]" is either an advanced undergraduate course or a graduate course.

DUDE you are an incredible content creator! Not only finding all the resources that best help newbies into classic maths, but also providing insight into their way of teaching!

I'm extremely overjoyed to have found this video. I've been wanting to self-study pure maths (with some physics as a little treat) for a year or two now, and am currently strengthening my foundational understanding of the basics, but was bumbling through the undergrad stuff with the help of the MIT courses and Stack Overflow. I'm incredibly grateful that you included videos alongside the textbooks, since I tend to learn better with a combination of both (but mostly the former). This video is like finding an oasis in the midst of a desert, thank you so much!