Understanding/Mastering Analysis in Topology, necessary? 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.
 A: 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.
A: 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.
A: 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.
A: 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.
A: 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.
