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I have spoken to one professor so far about this, which of course was helpful, and so I am looking for additional opinions: To work with topological tools that were built via analysis, should I be a "master" at that analysis? By this I mean, for instance, to use Seiberg-Witten Theory and Floer Homologies.
As an "entering" graduate student I am "purely" a pure topologist, as in I have no real training in analysis but Algebraic Topology under my belt for $\approx 6$ years. Now learning Seiberg-Witten Floer Homology and other Floer homologies, I tend to put all/most of the analysis (ex: compactness of moduli spaces) in a black box, and then continue to "learn". As a result, I am unsure if I am kind of wasting my time, i.e. if I can still utilize the theories effectively (and of course, I would like to extend theories). Is there a "good" balance between 1) simply accepting the analysis and 2) being able to do the analysis with both hands tied behind your back (as Kronheimer-Mrowka seem to do in their Monopoles and 3-Manifolds book)?

I am unsure how to make this question less vague / more precise, but I feel that there is a good underlying question here that can have an informative response.

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    $\begingroup$ I'm not sure I entirely understand the question: There are "manufacturers" who create tools (Floer, Kronheimer, Mrowka, Ozsvath, Szabo, Taub...), and there are "consumers" who use them. To be a consumer, you need no knowledge of the details of the construction at all- just calculate Floer homology groups and be happy. Whether you choose to be a manufacturer or a consumer, or a bit of both, is a personal decision, no? $\endgroup$
    – Daniel Moskovich
    Commented Mar 5, 2012 at 5:18
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    $\begingroup$ Is it though (personal decision)? The manufacturer surely had an applicational-goal in mind, no? And will the consumer's ignorance really get him far? And then there is, for instance, Taubes, who not only fits the manufacturer description, but also the consumer (with the Weinstein Conjecture, which doesn't seem void of detailed analysis). Sorry my questions are naive. $\endgroup$
    – Chris Gerig
    Commented Mar 5, 2012 at 5:50
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    $\begingroup$ It's hard to give generic advice about this kind of thing. On the one hand, knowing the hard technical details about how machinery works is very valuable. One on other hand, there's an infinite amount of stuff to learn and a finite amount of time to learn it. I've certainly seen graduate students and postdocs get bogged down in learning technical details to the detriment of their research. In the end, you need to trust your advisor to make sure you are striking the right balance. He or she knows what you need to be doing far better than people here do. Trust him/her! $\endgroup$
    – Andy Putman
    Commented Mar 5, 2012 at 5:54
  • $\begingroup$ I think complex/algebraic geometers have a similar issue in using the Hodge decomposition, which has purely algebraic consequences not all of which have a known purely algebraic proof, but without knowing how to prove the decomposition. $\endgroup$
    – David Corwin
    Commented Jun 15, 2014 at 17:46
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    $\begingroup$ Ran into a reputable well-connectrd Japanese topologist at a Starbucks in Japan once. Was teaching an undergraduate calculus class and was making notes on something about sin(x)/x but had no idea of its applications. The students are not well-served by a specialist in one field teaching outside his own unless he has striven for a more general competency. $\endgroup$
    – Tom Copeland
    Commented Dec 5, 2019 at 16:41

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I am very sad. We wrote "Monopoles and Three Manifolds" with the idea that a good graduate student who had read something like Warner's book (through the chapter on Hodge theory) could reasonably read much of the book. Oh well.

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    $\begingroup$ I will take this response as the lower bound of what I should be comfortable with then! I shall not disappoint. $\endgroup$
    – Chris Gerig
    Commented Mar 5, 2012 at 17:09
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In the long run, you will probably have to learn math on a "need to know" basis and not waste time learning technicalities that you don't really need to deal with in your own work.

On the other hand, you will never have as much time available to learn mathematics as now, while you are a graduate student, and you should take advantage of that. This is the time to learn as much math as you can, even if you're not sure you're going to need it. But you still need to choose what to learn and what to treat as a "black box". Get guidance from others, think about this yourself, and then just plunge in. I particularly liked doing working seminars with other graduate students with similar interests. This often led me to learn stuff that I wasn't initially interested in.

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You do not need to know analysis at the same level as the people that created these theories, but you need to have at least some general understanding and awareness of what goes inside, and of the possible traps.

To give you an analogy you might relate to, think how far you would get by learning singular homology axiomatically, with no understanding of what really goes inside.

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    $\begingroup$ To be honest, I think that most algebraic topologists use singular homology without ever thinking about the free abelian group on all singular simplices. Indeed, it is the axiomatic properties of the homology functor that make it computable, not its definition. $\endgroup$
    – Craig Westerland
    Commented Mar 6, 2012 at 10:52
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    $\begingroup$ I am certainly not one of them and maybe that is why I call myself a geometer. I find it hard to imagine discovering new geometric results using only axioms disconnected from intuition. To quote Bertrand Russel, "the axiomatic method has many advantages, similar to the advantages of stealing over honest work." $\endgroup$
    – Liviu Nicolaescu
    Commented Mar 6, 2012 at 11:15
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    $\begingroup$ I certainly didn't mean to impugn geometric intuition. Do you find the actual definition of singular homology amenable to your geometric intuition? That said, I think that the axiomatic approach to singular homology does in fact appeal to intuition, but it is largely algebraic intuition, with some very basic geometry thrown in. $\endgroup$
    – Craig Westerland
    Commented Mar 6, 2012 at 11:36
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    $\begingroup$ I am referring to the original idea of simplicial homology going back to Poincare. This is how I think about homology. There is certainly an important place in mathematics for formalization. In my mind the emphasis in the term algebraic topology should be on topology. This is not a judgement of value, merely personal taste. $\endgroup$
    – Liviu Nicolaescu
    Commented Mar 6, 2012 at 12:21
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I started the same way as a topologist and ignored all analysis that was not needed for my research project. Now as a post-doc, people don't really believe me when I say I don't like analysis, because they read what I did and it turned out to be pretty analytic.

So I would say, it depends on where you want to go. For me, the best (and probably only) way to learn analysis is by doing it. So as way said before: it depends on what your research is going to focus on, how much you need to know.

Yes, it will slow you down in research, if you have to learn the analytical tools as you need them. But getting a big toolbox and then only using a little screwdriver is completely over the top. So I would make sure you really know whatever you need to use in your own work and treat other things you read as a black box. You will find that your toolbox gets bigger and bigger over time. But there will always be black boxes around as there is much more stuff to learn than you can do if you want to have time for your own work.

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I was working with some fellow grad students (studying algebraic topology) a few years ago and they were having trouble computing some integrals. This was not unusual, but then they asked me if one ever uses integrals in research. I was a little shocked at first but then I realized they had never taken a rigorous analysis course. The conversation ended on a happy note, however, as we ended up discussing the similarities of a sigma algebra and a topological space.

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