[artagnon]

I've self-taught myself undergraduate-level Mathematics, and am currently enrolled in an advanced Masters program specialized in higher category theory at Université de Paris. Here are some thoughts:

First, the amount I learnt on a day-to-day basis from self-studying in my free time pales in comparison to the amount I learnt from a structured course ending with an exam, in which you have to commit to studying 2~4 hours of math every single day.

Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks. The problem sheets they distribute in the tutorials are very helpful, as is the discussion of problems that some students might have got stuck in.

Graduate-level math textbooks are fantastic, and the authors have poured 10+ years of their lives writing them. No online course with cute interactive video can substitute them. Having said that, there are some video lectures that can act as good motivation for studying a subject. For instance, Wildberger's lectures on algebraic topology (find them on YouTube) are a great starting point for the subject.

You first have to decide on what you want to learn, by looking through various fields, and narrow your scope. Mathematics is a huge discipline, and you have to decide what you like, and work through the prerequisites in a disciplined manner.

I cannot emphasize the importance of working through exercises enough. All mathematics textbooks have exercises at the end of each section, a subset of which you must work through.

Having said all this, you might find the following site interesting: it contains a lot of unpolished notes from studying various math textbooks, but it requires a lot more work to be useful to a larger audience. Look through it, and pick something you like. On request, I might find the time to add some material.

https://artagnon.com

[ege_erdogan]

I always appreciate when someone leaves the well-defined path and does something different, such as you self-studying Math and getting enrolled in a Masters program.

I am just curious about your process of actually joining the masters program. They usually require 2-3 letters of recommendation etc. How did you handle such requirements?

> Second, classroom lectures mainly act as a structured table-of-contents for working through various textbooks.

On a side note, actually studying math as an undergrad, I can't stress enough how much I agree with this, especially in a remote setup.

[artagnon]

See the other reply on the thread: https://news.ycombinator.com/item?id=24900707

[hnthwaccount]

Can you speak towards what you self-studied to get to an undergraduate understanding of mathematics and what resources, or books you used in the process? I’m also interested in the resources you used or recommend for learning category theory.

[artagnon]

Motivation and direction are important when starting out. I decided pretty early on that I wanted to be an algebraist, but would have to build some mathematical maturity before I could get there, so I had a rather shallow goal in the beginning: to be able to solve the previous years' math GRE papers. Off the top of my head, these were some of the books that I worked through:

1. Spivak's Calculus.

2. Johnstone's Notes on Set Theory and Logic.

3. Gamelin's Complex Analysis.

4. Hoffman & Kunz' Linear Algebra.

5. Dummit & Foote's Abstract Algebra; just the group theory.

6. Munkres' Topology; just the general topology.

Once I was happy with my preparation, I strived for a deeper understanding of Group Theory. I bumbled through Herstein, but didn't understand it very well. Then, I stumbled upon Artin's book, and worked through it using the outline provided on the MIT OCW course page, and I could confidently solve most of the exercises.

For category theory, the top resources that I would recommend are:

1. Mileweski's Category Theory for Programmers, the video lecture series. As is always the case with video lectures, this one can help motivate a Haskell programmer uninitiated in category theory.

2. Goldblatt's Topoi. It's fairly dated, but teaches category theory well, via its application to topoi.

3. MacLane's CatWork. I'm not especially fond of this one, but it's necessary to work through it.

The most important thing to understand when learning category theory is that it cannot be learnt in a vacuum: the subject is entirely vacuous, and you need to use it in other mathematical disciplines to give it meaning.

Good luck.

[jkmcf]

How long did you self study? Total years and average hours per day?

Merci!

[artagnon]

I started in late 2016. Initially, it was just during the weekends. I was a regular at the local coffee shop near my place in Boston: I'd go in with my iPad and some sheets of paper, spend 4~5 hours studying, while constantly consuming coffee. Then, I found a friend who had done a Bachelors in math, and we'd visit the local community college 3~4 times a week, and discuss general math and GRE problems on the blackboard for a couple of hours after work: it was a lot of fun.

In mid-2017, I moved to California and spent ~2 hours a day studying by myself, while maintaining a day job writing Haskell and Coq. Then, in late-2019, I quit my job and moved to India to study mathematics full-time (read: 4~6 hours every single day). I started meeting a professor at the local university once a week to discuss my solutions to problems in Miles Reid. We worked through Reid together. I also audited a course in algebraic topology at the university, simultaneously.

In mid-2019, I moved to Paris to continue studying part-time (1~2 hours a day) while working on a Coq project. After some shuffles, I found a professor I really liked, and we started working together. He wrote me the primary recommendation for the Masters program.

I hope the elaboration was more useful than two numbers.

[mangamadaiyan]

If I might ask - which local university was it that let you audit their course? This is not a usual occurrence in India, which is why I ask -- I'm extremely happy to hear that things like this even happen as outliers, btw.

The fact that it was a graduate-level course in algebraic topology seems to imply that the university in question is not a run-of-the-mill university, which makes me all the more happy.

[artagnon]

Chennai Mathematical Institute.

[mangamadaiyan]

That is simply wonderful! Good for you, and good for them!

[zrkrlc]

Was your undergrad GPA ever a factor in the admissions process?

[artagnon]

No, because my undergrad wasn't in math.