15
$\begingroup$

Is it necessary for one to understand analysis in order to pursue a career in mathematics?

Basically, I am very weak at analysis. But the problem is that most of the topics listed in the syllabus include analysis:

  1. Real analysis

  2. Complex analysis

  3. Topology

  4. Functional analysis

  5. Measure theory

  6. Manifold theory

All require a good understanding of analysis.

Is there any scope for me to survive in the mathematical world? I am only interested in Group Theory and Ring Theory (i.e, Abstract Algebra).

Is it possible for me to just carry on, or will I also have to develop a liking for the topics listed above? Is it possible to pursue a career without them?

Sorry if the question is very personal, but I don’t have other options.

$\endgroup$
  • $\begingroup$ When you say syllabus, what exactly do you mean, a course? Program of study for a degree? Some kind of (group of) exam(s)? $\endgroup$ – pjs36 Apr 7 '15 at 6:06
  • $\begingroup$ A course in mathematics $\endgroup$ – Learnmore Apr 7 '15 at 6:25
  • $\begingroup$ amazon.com/Primer-Infinitesimal-Analysis-John-Bell/dp/… $\endgroup$ – user117644 Apr 7 '15 at 6:26
  • $\begingroup$ Can you elaborate what is this book about @mistermarko $\endgroup$ – Learnmore Apr 7 '15 at 6:31
  • 7
    $\begingroup$ Necessary for what? $\endgroup$ – Qiaochu Yuan Apr 7 '15 at 6:50
33
$\begingroup$

Eric's answer is quite thorough, but it is missing an important point: in mathematics, everything tends to be related and intertwined. You might think that doing abstract algebra will get you far from analysis, but you would need to restrict your scope a lot more to do that, as there are many things relating the two, just to name a few:

  1. Group theory and group representation theory, which is very closely tied to harmonic analysis.
  2. Complex analysis is often useful whenever we deal with algebraically closed fields, both as a tool and as an illustration. Similarly with real analysis and real closed fields.
  3. Algebraic geometry is closely tied to analytic geometry.
  4. Topology (which you seem to include within analysis) is ubiquitous in mathematics, and that certainly does include abstract algebra.

I could probably name a few more, and I am neither an algebraist nor an analyst (though to be fair, I have had some education in both, quite a bit beyond the undergraduate courses).

There are branches of mathematics that could possibly get you further from analysis (some branches of mathematical logic, perhaps). That said, I can barely imagine how one can do serious mathematics without at least a modicum of topology.

Last but not least, in my opinion, really beautiful mathematics is the kind that (among other things) connects seemingly unrelated concepts and results in a way that is both unexpected and effective. By judging parts of mathematics as entirely uninteresting (especially such large parts as you seem to be indicating!), you severely limit your ability to even understand that and perceive that beauty (leaving you with a black box at best), not to mention making anything like that on your own.


Edit: I've just recalled a funny bit of somewhat related trivia: in the first chapter of Serge Lang's algebra (which is a canonical reference for abstract algebra, from basics up to quite advanced topics, and generally steers clear of analysis), Lang gives two concrete examples of monoids. The first one consists of the natural numbers with addition. The second one consists of... homeomorphism classes of compact connected surfaces with connected sum, indirectly referencing surgery theory and classification of compact surfaces.

$\endgroup$
  • $\begingroup$ ok I will definitely try to understand analysis now;but are there any interesting /good books about the topics listed above that can make me feel comfortable $\endgroup$ – Learnmore Apr 7 '15 at 15:13
  • $\begingroup$ I mean books that provide a good link between algebra and analysis $\endgroup$ – Learnmore Apr 7 '15 at 15:15
  • $\begingroup$ @learnmore You might take a course in manifolds. In manifold theory you get a combination of ideas from analysis, linear algebra, and abstract algebra. The book "An Introduction to Manifolds" by Loring Tu is great. $\endgroup$ – Daniel Watkins Apr 7 '15 at 15:35
  • 2
    $\begingroup$ Hm, I think I was trying to get at this idea, but I'm glad that you brought it out explicitly and gave some good examples :) $\endgroup$ – Eric Stucky Apr 7 '15 at 20:36
  • $\begingroup$ @learnmore: I don't think I can give you anything like that. Firstly because I have learned basics before I learned about connections between things, and secondly because I tend to study from my notes and only use textbooks for reference. That said, I highly recommend Serre's "Linear representations of finite groups". I've found the book very useful, and easy to read. I vaguely recall some claims that the book was written for non-mathematicians. $\endgroup$ – tomasz Apr 7 '15 at 20:53
38
$\begingroup$

To me, asking if you need to understand analysis is roughly* like asking "Is it necessary for one to understand how to operate a computer to pursue a career in mathematics?", in that the answer is technically no, but

  • Everyone else does
  • They'll assume that you do too
  • There's no good reason not to know
  • By not knowing, you are making things incredibly difficult on yourself
  • By not knowing, you are making things incredibly difficult on your collaborators and peers
  • If you knew, you would probably live a better/easier life in general
  • You probably didn't get good enough at it with your free sample to make an informed decision about whether you like it or not
  • Given the choice between knowing and not knowing, I can't imagine why you would pick not knowing
  • A stubborn insistence to go on without it is probably indicative of deeper problems that will hamper your progress.

(* to zeroth-order approximation)

$\endgroup$
  • 6
    $\begingroup$ (Practically speaking, you will need to pass a preliminary exam in some sort of analysis to get a PhD in pure math from a reputable university, even if you only ever do algebra.) $\endgroup$ – Eric Stucky Apr 7 '15 at 7:20
  • $\begingroup$ @learnmore: For completely random reasons, I came across a book called "Metric Spaces" by Micheál O'Searcoid. It looks to be a fairly gentle introduction to the early parts of baby Rudin, and you might find it helpful. $\endgroup$ – Eric Stucky Apr 12 '15 at 13:40
  • $\begingroup$ that was extremely helpful $\endgroup$ – Learnmore Apr 13 '15 at 2:02
18
$\begingroup$

Yes.

Analysis, as far as you have seen it, is training on mathematical reasoning, on the meaning and usage of quantifiers, on organizing more or less complex proofs, and such. You will not get far without all of this.

$\endgroup$
  • 5
    $\begingroup$ One could add: Necessary, but not sufficient $\endgroup$ – Hagen von Eitzen Apr 7 '15 at 11:41
  • 9
    $\begingroup$ -1 Because the answer is "immensely unmotivated" (to use your own words). $\endgroup$ – Bill Dubuque Apr 7 '15 at 17:47
0
$\begingroup$

Well, considering the fact that the limits of sequences of discrete objects - and one can very well define limits of graphs, of logical models, of sets and power sets of stuff, of groups and rings so on - are typically continuous if they have meaningful definitions and if they exist at all, then yes, you'll most probably need to have a reasonable command of analysis in order to pursue research in Abstract Algebra.

9 years ago, my advisor (Prof. Eldar Fischer) co-wrote a paper in combinatorial property testing, completely discrete basically:

A combinatorial characterization of the testable graph properties: it’s all about regularity

and strangely enough, in the same conference, another paper contained just about the same result - derived through analytic and measure-theoretic tools:

Graph limits and parameter testing

It was quite the coincidence!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.