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.



Understanding/Mastering Analysis in Topology, necessary?
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.


