After drinking too much coffee while solving the NYT crossword puzzle (I can always finish the NTY Xword up through Thursday, but Friday and Saturday continue to be the bane of my existence!) I started to think about Brouwer's Fixed Point Theorem (BFPT). As a topologist, I think of BFPT as
Every map from a disk to itself has a fixed point.
What does this have to do with coffee?!? If you think of the surface of the coffee in the mug as the 2-disk (assuming that you drink your coffee out of a circular mug) then BFPT can be interpreted as saying
If your coffee is at rest, and you stir it with a spoon, once it comes back to rest there is at least one drop of coffee that is exactly where it started before you stirred!
There is a more general statement that implies BFPT:
Every map from a convex compact subset of a Banach space to itself has a fixed point.
Banach is an interesting fellow to talk about, but I'll save that for another post. It's Brouwer that I'd like to focus on for a moment. BFPT has so many well-known proofs! The Wikipedia link above outlines (a modest) 8 proofs, although I'm not sure how I feel about "Reverse Mathematics". In the beginning (as "legend" has it) I understand that Brouwer himself was quite upset that the original proof of BFPT proceeded by contradiction, as opposed to being a constructive argument. As far as I'm concerned, a proof is a proof! Brouwer's notion of intuitionism is, well, a little bit zany for my taste. But then again, it's hard to argue with a man whose name appears on such a prevalent theorem in mathematics.
I'm now off to shittily wrap presents. Wrapping presents is much harder than math.
--A.M.M
