*Try to learn something about everything and everything about something.* (Thomas Huxley)

Maths phobia is a pervasive problem in the wider community. Unfortunately, it sometimes also exists among professional mathematicians (together with its distant cousin, maths snobbery).

If it turns out that in order to make progress on your problem, you have to learn some external piece of mathematics, this is a *good thing* – your own mathematical range will increase, you will have acquired some new tools, and your work will become more interesting, both to people in your field and also to people in the external field.

If an area of mathematics has a lot of activity in it, it is usually worth learning why it is so interesting, what kind of problems people try to work on there, and what are the “cool” or surprising insights, phenomena, results that that field has generated. (See also my discussion on what good mathematics is.) That way if you encounter a similar problem, obstruction, or phenomenon in your own work, you know where to turn for the resolution.

One good way to learn things outside your field is by attending talks and conferences outside your field.

See also “Learn and relearn your field“.

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28 December, 2007 at 3:11 am

anonymousDear Professor Tao,

Do you have any advice on how to stay sharp with so many different areas of math (or if it is even necessary)? I find it is hard enough to be an expert in a small niche in just a single field!

More specifically, I think it makes sense to learn areas outside one’s field insofar as having a rough outline of some of the key ideas, or being able to recognize a theorem and knowing where to look for more details. In your case, you’ve endeavored to do much more: you are constantly writing (be it research or expository blog articles) in many disparate areas. When you switch from topic to topic, are they all fresh on your mind? (For me personally, I often use Wikipedia as a refresher :-)

28 December, 2007 at 12:35 pm

Terence TaoDear anonymous,

I find in fact that I learn areas outside of my field (or within my field)

becauseof efforts to write expository articles, teach classes, etc., more than the other way around; these articles I write for this blog are as much for my own edification as for anyone else’s. I started writing articles like this several years ago (mostly for myself, although I do keep quite a few of them on my own web page), after several frustrating experiences in which some neat idea or result I learned in a talk or paper managed to disappear from my memory to the point that I could not easily reconstruct it. The act of writing one of these things forces one to really get the whole story straight in one’s own mind; otherwise it’s too tempting to just be content with some superficial understanding of some mathematical topic. (Of course,afterI write the article, my understanding tends to revert back to a more superficial level as the memory fades, if I don’t use the material regularly, but the difference is that now I have a written record to refresh myself whenever necessary.)It also helps to have some immediate motivation for trying to understand something, for instance if it is connected to something else one is interested in. I doubt I could write anything particularly insightful about, say, the local Langlands correspondence; it’s great mathematics, but at present I don’t see any connections between it and what I am presently working on.

And yes, I rely heavily on Wikipedia (and Google, and Mathscinet, and the web pages of various mathematicians) every time I write one of these things :-) .

15 June, 2008 at 3:00 pm

这等牛人也在wordpress上写blog！ « Just For Fun[…] useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And […]

10 July, 2009 at 1:24 pm

QwertyDear Prof. Tao,

I am at graduate level (not Ph.D yet, hopefully I’ll start that next year) doing abstract algebraic geometry. I do not know any Fourier analysis and I am wondering if I should learn some or not. What would you recommend? Should someone like me take the time and learn it (for some reason) or am I better of spending the time doing SGA (say)?

11 July, 2009 at 2:27 pm

Terence TaoWell, Fourier analysis is a fairly basic mathematical tool, and worth learning at some point, but it is fairly disjoint from algebraic geometry, and so you might not be able to connect the material with what you are doing. It may be better to spend time learning material which is more adjacent to your area first, and then branch out once you have some handle on those adjacent fields. (For instance, from algebraic geometry, one has a foot in the door to understand algebraic groups; from algebraic groups, one can step over to representation theory; and from representation theory, one can jump to Fourier analysis. There are of course many other routes available.)

There is some discussion over at

https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/

as to the how much of EGA and SGA one should be reading for algebraic geometry.

23 December, 2009 at 6:28 pm

If you want to go beyond the Princeton Companion to Mathematics then the Oxford User’s Guide to Mathematics could be an answer « Successful Researcher[…] deserved. But where should an interested student (or even a professional mathematician who wants to extend her or his professional range, for that matter) go in order to deepen the knowledge acquired from PCM without getting bogged down […]

23 March, 2010 at 9:07 am

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7 August, 2010 at 11:20 am

Tinh Quoc BuiHello Terence, … I am so impressed with these nice advised words, many thanks, they are absolutely useful…!

28 September, 2010 at 8:55 pm

Rob TDear Terry,

Do you think you will ever teach an undergrad abstract algebra course?

17 July, 2011 at 8:17 pm

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10 November, 2011 at 4:32 am

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24 December, 2011 at 10:22 pm

AnonymousMy main area of research (PhD, subsequent 5 published papers in 2 peer-reviewed math journals) is/ has been differential algebra. I find I still can swim in this huge ocean without ever making landfall onto other branches of mathematics, if I want. Wikipedia has been a great way for me to learn what I should have learned in graduate school the first time – such as category theory, homology sequences – as well as math “for fun” (as I see it) – such as math the physics of quantum mechanics and general relativity.

Don’t know if you’ve heard this one before from someone heavily into differential equations as I am, but, the other branch of math with which I have a great passion to push my abilities is game theory combined with formal logic applied to law and justice. Modeling law, justice (fairness and unfairness) as sequences of logical “P implies Q” like statements, with given players, to determine logical inconsistencies in human thought, is a raging obsession of mind. It returns me to my roots as an engineer – mathematical modelling the difficult, the vague, the poorly defined, and the emotionally sensitive – e.g. quantifying quality of life, amount of suffering and sacrifice, value of work, etc.

But, I never pursued the widely disparate fields of differential algebra and modeling justice with formal logic and game theory for the purpose of getting a job in these fields. I pursued them after a prior life-changing epiphany, brought on by my engineering courses, and work in a research lab, as well as direct political interaction with other human beings,

as an absolute necessity, not only for my own survival, but for the future of the world. Frankly, I do not understand why most people who chose to get jobs as professional mathematicians do not see their mathematics the same way.

21 May, 2013 at 3:43 am

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19 July, 2013 at 8:57 am

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6 February, 2015 at 5:27 pm

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11 July, 2015 at 6:00 am

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