Wednesday, March 21, 2012

"simple" still means simple, but ..

finally: back in helsinki!

*sigh of relief*

i don't care if grοmov himself invites me .. to new york, france, anywhere;
i solemnly vow to stay put for at least one month!
last week i helped a friend stay put;
it's a lot easier than helping someone move.

i just went over to his house and made sure that he did not start to load all his stuff into a truck ..

- mitch hedberg.

to "celebrate" this occasion, i'm making corrections to a particular manuscript that i've set aside for close to 6 months.  in one part, i had written that:
"The proof of Lemma 4.3 is technical but the idea is simple."
i re-read the proof, then frowned: "wait: why is this simple" ..?

then i remembered: i drew a picture, felt satisfied,
decided to rewrite the statement as:
"The idea of Lemma 4.3 is simple, but the proof is rather technical" ..
.. and added another clarifying sentence or two.

to be fair, if you're not used to weak topolοgies, then the strategy doesn't look like it should work at all:
the point is that weakly-convergent yet geometric approximations of metric spaces give rise to isomorphisms of certain generalised differential operators (called derivations);

in the case of manifolds, this corresponds to how a diffeomorphism gives rise to a push-forward map between tangent bundles.

in the generality of metric spaces, though, the really cool part is that you can do this with certain embeddings. the bundles may degenerate, but the dimension won't!

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