How to Learn Advanced Mathematics Without Heading to University

10 years ago I was an early employee of a risk analytics firm. I got pushed into a quant roll because I had an engineering background and could solve PDE's.

My first meeting I remember the head of the quant team hand-waved over a bunch of math that took me two weeks to go through on my own.

I had to spend 2-3 nights a week for 1 year going over my old statistics texts, Differential equations text and Knuth's Concrete Mathematics before I moved onto Stochastic Calculus.

It would have been almost impossible without having a team of people who were willing to help me get through sticking points. That was 8-10 years ago. I still spend atleast 1 night a week learning new math, though now its mostly non-linear regression and chaos theory.

And to be honest I still get alot of quants looking down on me, "A guy who dropped out of his masters, trying to claim he's our equal?". That does bother me a bit, and it means I probably couldn't get a quant job at Goldman, but I'm very proud of where I am now and how I got there!

TL/DR It can be done, but it gets done over the course of years, not months or weeks. You won't get there without applying what you learned to real world applications, reading the text's aren't enough.

It's not quite entirely by yourself, as in the webpage linked, but it worked like this:

  1) Get sent problem sheets and a list of what chapters
     in a textbook to work through.
  2) Read textbook.
  3) Solve the problems in the textbook.
  4) Watch a one-hour live webcast to you (and a dozen 
     other people) four times.
  5) Send off completed problem sheets; if you make the 
     pass mark, your prize is a seat in the exam.
  6) Optionally pay for a weekend of direct instruction to 
     you (and a dozen others) in person somewhere (I did
     this 3 times out of six, I think).
  7) Sit three hour exam in supervised exam hall.
  8) Go home, open next textbook.
  9) Repeat four more times, and then write a dissertation.

Also not the same; a formal, full-on three hours exam in a formal exam hall. This sets the bar pretty high and is a really good indicator to yourself. I think this makes a real difference; someone else validating that yes, you really do know what you're talking about (or not, as the case may be - I think that of everyone who starts this, less than fifty percent pass the first exam, and if I had to guess, I'd say that it's not so much that the maths is hard; making yourself study it to the level required month after month is hard).

I estimated that the time I spent on this was roughly equivalent to working full-time for 6-8 weeks per year. Interestingly, if I'd been given an actual 8 week block, I don't think I could have done it. I couldn't spend 8 hours a day reading a maths textbook and get the same from it as if I'd done it in four two-hour blocks spread over a week.

Coming back from an exam, sitting down at the table, putting all the notes and papers and books from the previous subject to one side and then sliding the virginal textbook and brand new notes and problem sheets onto the table was brutal. It can be done, but you have to want it. I found that in terms of taking in new knowledge, I couldn't do more than a couple of hours with the textbook at a time. For practising and solving problems, I could sometimes sit down at lunchtime on Saturday to just have a go at one quickly and stand up again eight hours later, table covered in notes and the problem at hand having been thoroughly explored and answered (typically, sadly, spread over several pieces of paper which I would come back to in the future and condense into a smaller paper space).

The study habit it forced into me is enormously valuable in itself. I can now pick up a high-level maths text book and simply start learning. I won't understand much of it, but I know that if I keep at it, I will. Grind, grind, grind. Not just mathematics, either. Someone gave me a Learn Japanese book for Christmas and now it's in me. I ground through it, and now I've got another one and I'm going through that. It's as if the subject doesn't really matter, so much as the studying.

> Nope

1. "good experience manipulating differentiable functions and their derivatives" sounds weird in prose.

2. Some continuous functions are differentiable. Those ones have derivatives you can manipulate. In fact knowing when a function is not differentiable is a pretty useful skill.

> The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing.

Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity".

> only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory

Probably true. Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges. Fact is most people in industry are looking for "calculus for engineers" levels of formal understanding. Enough to be useful and make money while avoiding expensive mistakes.

> and were looking to hire people like themselves.

Also probably true. This is a good assumption not just on Wall St but everywhere.

", chances are if you are considering studying advanced mathematics you will already have formal qualifications in the basics, particularly the mathematics learnt in junior and senior highschool (GCSE and A-Level for those of us in the UK!). "

The author is not a Fields Medalist writing to other PhD mathematics graduate students. In such a case, the adjective "advanced" would have a different meaning.