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I've come to like linear algebra far more than analysis and was wondering how much analysis one would be required to learn in order to pursue a career in research in an algebraic field. For example, at the graduate level, if I were interested in focusing on algebraic topology or algebraic geometry then would learning the contents of analysis 2, real analysis, and differential equations be enough before one exclusively focuses on algebra core subjects? Is it common for mathematicians of one field to be very weak in other fields? Like somebody who does exceptional research in algebraic geometry who would struggle with solving problems in a complex analysis text?

Thanks in advance!

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    $\begingroup$ Although it has the preferences switched, you might be interested in this question : math.stackexchange.com/questions/355418/… $\endgroup$ – Arnaud D. Nov 29 '17 at 13:48
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    $\begingroup$ Is it common for mathematicians of one field to be very weak in other fields? A mathematician is necessarily weak in most fields because of the variety and depth of mathematics that exists now. Of course, it is a boon to be as talented as you can be in multiple fields. Typically the fields do "synergize" well. For me, algebra, geometry and topology is the raft of fields I drift on. On the other hand, I am totally clueless about something like Ramsey theory or infinitary combinatorics, or research level statistics. $\endgroup$ – rschwieb Nov 29 '17 at 14:10
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    $\begingroup$ Nobody can really dictate your interests to you, though. Out of necessity you'll probably have to pick up fields that border on your field. And if another field (any field) intrigues you, there's nothing stopping you from picking it up. The academic community benefits from this sort of variance. $\endgroup$ – rschwieb Nov 29 '17 at 14:11
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    $\begingroup$ One thing to keep in mind is that you should continue to learn new math for the rest of your life. As you do research, you may come to a point where you need to study a new topic a bit in order to incorporate its methods into your work. Also, you should strive to have colleagues with distinct expertise. You only need to be an expert is a small branch, but you should be roughly familiar with a lot more, basically the standard graduate curriculum add others have elaborated on. $\endgroup$ – jdods Nov 29 '17 at 14:51
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    $\begingroup$ Keep in mind that linear algebra is (at the level taught in a first course) very basic, and more advanced/abstract algebra may feel very different. And even analysts need to know linear algebra; some of them even love it, yet still prefer analysis to algebra more generally. $\endgroup$ – Toby Bartels Nov 29 '17 at 22:56
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At the graduate level you'll have to know standard (advanced) textbook material in both algebra and analysis to pass your qualifying exams. You shouldn't have to struggle too hard with the problems.

When you start focusing on your own research your abilities in the unrelated fields will get rusty. Years later the then current graduate curriculum might give you trouble unless you went back to study it anew.

It's always a good idea to keep somewhat current in several areas. Some of the major outstanding problems (e.g. the Langland's program ) are precisely about how different fields shed light on one another.

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There is no good answer about what is "enough". But I can say two things:

  • Making a "specialization decision" very early is definitely a bad idea. Basically you are deciding that you don't like/don't care about/don't need topics you know nothing about.

  • Even if you make the decision to specialize, you cannot predict years in advance what tools or ideas might be useful to you. Many of the most significant advances in math, either in a global sense or in a personal sense (as in one's own research) are often fueled by knowledge of other areas. The more math you know, the more ideas available to you.

Final thought: every top mathematician I ever met had a broad knowledge, way beyond their specialization.

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There are some subjects that are required and occasionally mandatory to study in a graduate program. A graduate level course in Complex Analysis, Real Analysis and Algebraic Topology, Abstract Algebra are usually mandatory. Before focusing on a specific area, even if you decide about this very early, you need to cover certain graduate-level coursework in any case.

Linear algebra appears in many areas e.g. Differential Geometry, and is a subject required to know well enough for pursuing graduate studies. So enjoying this subject shouldn't imply that you should pursue one area or another.

That is why before starting research or focusing on a certain area you either pursue a master's degree to cover material in certain areas and then enter a PhD program or go through a 1-2 years preparation within a graduate program in order to learn and finalise your decision.

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The other answers have made all the important general points, IMO, so I will add a couple of things from my own experience.

Like you, my tastes favor the algebraic end of the spectrum, plus (geometric and algebraic) topology; I never much cared for real analysis.

When I encountered complex analysis, it was a revelation: this stuff is beautiful. Totally different feel from real analysis.

Functional analysis is a blend of linear algebra and analysis, and while I don't rank it quite as high in my own personal mathematical beauty contest, I would be sorry to have missed out on it.

The moral has already been stated by Argerami: "Basically you are deciding that you don't like/don't care about/don't need topics you know nothing about."

Finally, since you like linear algebra, take a look at Hoffman & Kunze sec. 6.4 for a nice application of it to the theory of ordinary differential equations.

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First of all, I edited your question tags and tagged your question as soft question. I hope that you wouldn't mind it.

I'm an undergraduate student, therefore I'm probably not qualified to give you advice for your future career path. But as far as I'm aware, most of my friends who are now in graduate level, no matter how good they were at the undergraduate level, wish they had spent more time learning the concepts taught in undergraduate level.

As your question about whether one can become a mathematician in one field without knowing the other fields well, I can give you a cool example of an elementary result in commutative algebra that is related to analysis. There's a famous lemma in commutative algebra that's called prime avoidance lemma.

Even though it's a result in commutative algebra and it's proved in the finite case in a commutative algebra course, the result can be extended to a countable union which shows a nice interaction between algebra and analysis.

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  • $\begingroup$ Could you maybe elaborate further on what the prime avoidance lemma has to do with analysis? $\endgroup$ – Stefan Perko Apr 7 '18 at 12:55
  • $\begingroup$ @StefanPerko: for further information, please refer to the article Sharp, R. Y.; V ́ amos, P. Baire’s category theorem and prime avoidance in complete local rings. Arch. Math. (Basel) 44 (1985), no. 3, 243–248 $\endgroup$ – stressed out Apr 7 '18 at 21:25
  • $\begingroup$ Thanks. So tl;dr: Baire's category theorem is used to prove the prime avoidance lemma and related propositions. $\endgroup$ – Stefan Perko Apr 8 '18 at 6:53

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