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.