Ask HN: How to learn advanced maths | Hacker News Reader

Ask HN: How to learn advanced maths

15 pointsdorchadas7 comments

I studied physics as my undergraduate degree and, as such, only really studied up to differential equations in a pure maths context. However, since graduating, I've been working mostly as a maths teacher, and it's made me much more interested in learning more about advanced math, outside the basic geometry, algebra and calculus we teach. It's also made me much more interested in learning about rigorous mathematical proofs, especially in a research framework.

Does anyone have any good textbooks or resources to help self-educate myself on more advanced maths? I still have my linear algebra and diff. eqs books that I'll go through again, but I'd love other recommendations. I've found the post series on Quantstart [1], which looks like it was never completed, and I know HN has discussed them some before ([2] being one example) but was wondering about anything else you all might know about besides going back to school (which I seriously might do, to be honest; teaching it has made me really fall in love with maths again, and made me regret studying physics as opposed to maths!).

I do know this is vague, and mathematics is a huge field with lots of subbranches, so just any resource you'd like to recommend to any of those subbranches, or, perhaps, something of how an undergraduate curriculum would work leaning up to graduate level work? Thanks in advance!

[1]https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1

[2]https://news.ycombinator.com/item?id=11267456

ETA: I would much prefer materials that have solutions readily available. Since I am self-teaching, I want to be able to confirm my answers to problems without resorting to MathOverflow or other resources every time. Of course, I am not against materials without solutions, especially if they are the best materials available.

One thing you want is some equivalent of the "Foundations" course described at https://www.quantstart.com/articles/How-to-Learn-Advanced-Ma...

You want to make your way to the point where you can write a proof and check it without needing feedback telling you where it falls down.

In addition to that, you've got the basic undergrad math major curriculum.

Maybe look at http://web.evanchen.cc/napkin.html as some sort of goal, to understand everything you find interesting there, and a small set of medium to hard problems that you could use to track your progression. (It goes quick so I'm suggesting it as an example plot of a subset of a curriculum, not necessarily a great learning resource. It's clearly aimed toward a particular sort of reader, and it is a bit tilted towards parlor trick mathematics. It's kind of a neat document if you finished a math major a decade ago. Edit: this link is weak on analysis.)

Oh, this is definitely an answer you should read critically.

I would second going back to school if you can afford it. You can always start going part-time and then transition to full time if you really see it as another career path.

I'd also recommend the Art of Problem Solving's Calculus text, it has a solution book, as well as their Volume 2 book. Try working through those on your own, since you have great background in math you should find it quite enjoyable, and perhaps somewhat challenging.

I would recommend "Discrete mathematics demystified" by Steven G. Krantz. It is an introduction to topics that you may have never encountered before

I can't suggest brilliant.org highly enough.

Not a mathematicians, but I always got the impression that at the beginning it's all about doing it yourself...and later it's all about reading what the top minds are doing

I repeat my comment from https://news.ycombinator.com/item?id=11274264

„A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need. And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.“

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