[i started this last month, on 20 sept.
i still agree with what i wrote then.]
ALWAYS START EARLY. as someone who failed to do so, trust me: this task takes up a lot of time.
remember how difficult it was to write a research statement, when applying for postdoc jobs? remember when you had to be nontechnical and not span past 5 pages, in discussing your thesis work?
now add 10 more pages, and write more diplomatically: no full proofs, but some details so that the math makes sense. in short, you need 10 pages full of ideas -- not summaries of past work, but unfinished, new ideas.
i exaggerate, of course: unless your field is well-established, there is a good deal of exposition involved. i think half of what i wrote was to explain why the analysιs on metric spaces is even relevant to study, what it affords you and why it is hard.
"analysis on metric spaces is interesting in its own right" is one reason to study the area, but i hesitate to say that it's the only reason. besides, can't everyone say the same about their own field?
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