
Ask HN: How to learn advanced maths - dorchadas
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&#x27;ve been working mostly as a maths teacher, and it&#x27;s made me much more interested in learning more about advanced math, outside the basic geometry, algebra and calculus we teach. It&#x27;s also made me much more interested in learning about rigorous mathematical proofs, especially in a research framework.<p>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&#x27;ll go through again, but I&#x27;d love other recommendations. I&#x27;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!).<p>I do know this is vague, and mathematics is a huge field with lots of subbranches, so just any resource you&#x27;d like to recommend to <i>any</i> of those subbranches, or, perhaps, something of how an undergraduate curriculum would work leaning up to graduate level work? Thanks in advance!<p>[1]https:&#x2F;&#x2F;www.quantstart.com&#x2F;articles&#x2F;How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1<p>[2]https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=11267456<p>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.
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SamReidHughes
One thing you want is some equivalent of the "Foundations" course described at
[https://www.quantstart.com/articles/How-to-Learn-Advanced-
Ma...](https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-
Without-Heading-to-University-Part-1#year1-foundations)

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](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.

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HiroshiSan
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.

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

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extremum134
I can't suggest brilliant.org highly enough.

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Havoc
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

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amai
I repeat my comment from
[https://news.ycombinator.com/item?id=11274264](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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thedevindevops
3 blue 1 brown @YouTube

