Thoughts on Geometric Analysis, and on PDE.

[After a brief experiment with LiveJournal, I've decided that from now on, my mathematics posts will be put here. See comment above for personal rants.]

Certain things in life are inevitable. For instance, just when you think that the job is done, instead it resurfaces and you must tend to it once more. So it goes with the manuscript.

My collaborator has written back and informed me of a paper, where similar results (that I took some pains to prove) had been proven already. I suppose in a way this is actually a good thing which has happened; at the very least, I was very likely proving the right things, and now there is a reference to see if my method of proof could be right.

On a different note, a friend of mine (and fellow graduate student in analysis: see here) has made progress on a Uniformization Theorem in the metric measure space context.

(I'll omit the technical details; you can bug him about them instead)

This is a fine thing, indeed. It's an enviable result in alarming generality, much like other work in that area of study: "Geometric Analysis" and in particular, "Analysis on Metric Spaces."

I prefer to think of this as topology-driven analysis, which strikes me as quite hard: hard in the sense that the methods of proof can be at times quite axiomatic and definition-driven. The geometry becomes rather intangible and there is less visual intuition at your disposal; I can't seem to make heads or tails of it.

By contrast, potential theory seems to make perfect sense. PDEs make sense. Variational problems make sense. Occasionally I wonder if I'm in the right sub-field. I may have found an academic niche, but it may also be dangerous to become too comfortable in a single thing. I may lose my powers of creativity and flexibility in the process.

Already so much mathematics doesn't make any sense to me, and what little I know is enough to confuse me. It's frightening.

I really don't know if there is a point to all of these things that I write. That's the nature and the purpose of a blog, I suppose: to settle the malaise and uncertainty in thought, by releasing them into the world. But perhaps there's more to it than that.

Seeing what goes on in my department and what my peers have accomplished, it's quite easy to feel that my own efforts are made smaller and less in worth. It still seems like I haven't done anything of consequence and that it will remain so for a long time, and meanwhile I see great things happening, all around me ...