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I'm not asking about Linear Algebra that's doubtless vital to Finance. I deliberately haven't stipulated the kind of Quant job, as I'm asking in general. This post lists examples of Abstract Algebra. But ought you learn it on your own? Or ought you study them in university? I see mixed messages. bigbadwolf counsels

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity." The problem with the applied courses is that though they may be more "relevant," you're always at risk of having no real theoretical understanding and merely having mastered a cookbook of techniques with no conceptual focus. So ideally you want your numerical analysis and applied linear algebra to have a theoretical backbone, something that gives intellectual coherence to the subject matter.

Barny

Completely agree with bbw. Learn something that will force you to think in a different way about maths. Abstract algebra and analysis will do that. Complex analysis did that for me, too, so I'd recommend that too. If you want to be a quant you can certainly do too much pure maths, though. Once you've learnt how to think like a pure mathematician there is little utility in taking lots of pure maths options. Advanced courses in pure maths tend to be rote learning and regurgitation of proofs rather than concepts, so avoid that.

Yet bigbadwolf discourages it:

Abstract algebra is also great fun. But also, alas, of little or no applicability to finance.

Jae Sang Shim

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity."

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  • $\begingroup$ ." If you want to be a quant you can certainly do too much pure maths, though" how? $\endgroup$
    – Trajan
    Commented May 3, 2020 at 9:51
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    $\begingroup$ As usual, this is a classical utility maximisation problem: you have limited resources (time; courses; maybe energy) and want to reach a desired outcome (a quant position; also quant expertise?). So it really depends on how much time available you have (are you starting a bachelor? are you starting a one-year masters?). Personally, unless you come from a pure mathematical background and have already covered many other topics, I would put abstract algebra far down a list. $\endgroup$
    – Daneel Olivaw
    Commented May 3, 2020 at 10:24
  • $\begingroup$ First off, there is a lot you can study on stochastic calculus. Have you considered all what has been written on martingale theory, and how is that applicable to American option pricing problems? Analysis is also fundamental, in particular it's really useful in order to have a solid background on all the numerical analysis that is used in finance for tasks such as all the different types of optimisation routines used in model calibration. $\endgroup$
    – Daneel Olivaw
    Commented May 3, 2020 at 10:26
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    $\begingroup$ Personally, I have not studied abstract algebra to any meaningful degree, but I've never come across any paper or problem in quantitative finance where I've thought "damn I wished I had studied more abstract algebra..." $\endgroup$
    – Daneel Olivaw
    Commented May 3, 2020 at 10:29

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This doesn't directly answer your question, but I feel strongly about it and want to share my view and experience: you should concentrate on things you like and you are good at and put your best effort into it.

If HFT sounds cool to you, but you hate coding doing CS major won't help you. If pricing exotic derivatives sounds sexy, but stochastic calculus just doesn't make sense it probably won't work either.

Back to algebra - if you really enjoy doing it, then by all means go for it, do a math major, and it might help you in your quant career. But doing it just to get a quant job will be waste of your time.

Top quant funds will always prefer someone who's excellent at what he chose to work on (could be astrophysics, algebraic geometry, whatever) and the only way to get there is to choose a subject you're really really really into.

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  • $\begingroup$ This is excellent advice. $\endgroup$
    – Bob Jansen ♦
    Commented May 4, 2020 at 7:50
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In finance, mainly linear algebra is very useful and necessary to know. Algebra in general is not necessary but I would recommend learning at least basics. It would improve your logical reasoning and abstract thinking which is useful in any branch dealing with numbers, including finance.

Just give you an example, I have been amazed by hypercomplex numbers, the generalization of complex numbers. They are very different from numbers in $\mathbb{R}$ or $\mathbb{C}$ as they do not commute, eigenvalues of matrix composed of hypercomplex numbers are class of equivalence etc. There is no direct application in finance but they are interesting. However, to be precise, hypercomplex numbers are used in random matrix theory which has some application in finance.

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    $\begingroup$ Thanks. Just to clarify, I agree "linear algebra is very useful and necessary to know". I was asking just about Abstract. $\endgroup$
    – user31928
    Commented May 3, 2020 at 20:29
  • $\begingroup$ @Greek-Area51Proposal: OK, I though you are talking about general algebra, the linear one is part of. I expanded my answer, maybe it will be useful for you. $\endgroup$
    – Martin Vesely
    Commented May 3, 2020 at 21:01

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