Perhaps a half-dozen times in the past week, I’ve read sentences with contain the phrase “must needs.” I have never considered this construction before; frankly, it sounds totally bizarre to my inner ear (my spiritual inner ear, that is).

Thus, it must needs be that I’ve been teleported to another world, a world in which the English language developed differently than it did in the world from which I came. This tiny grammatical gem is the only evidence of my true origin.

I gave a seminar talk on

Herlihy, Maurice and Rajsbaum, Sergio. Algebraic topology and distributed computing–-a primer. 1995. 203–217. MR.

This paper doesn’t do it (but Rajsbaum’s MSRI talk did), but the result can be reformulated combinatorially, so that the algebraic topology appears as an instance of Sperner’s lemma; this is the sort of thing that should be done at mathcamp.

Here is something that amuses me, but I know that if anyone else said it, I would find it extraordinarily annoying: seeing as these results apply to anything (I mean, the local model of computation is irrelevent), this is an example of how deterministic systems, when combined with each other, yield non-deterministic results (though I have to be careful what I mean by “deterministic”—the system as a whole is determined, but non-deterministic from the perspective of the agents in that they cannot determine the outcome). Clearly I should write a philosophy paper, called “Free will and algebraic topology: a primer,” in which people are vertices in the simplicial complex of all possible worlds.

It will be better for all of us if I stop now.

I’m still trying to find (two!) new roommates (since my current roommate bought a place, and is moving out on December 15th). If you know anybody who would like to move in with me, I’d love to know about it.

There are some pictures of my home.

I wonder if anyone knows about alphabet songs in other languages? I’d be particularly interested in knowing about Greek and Hebrew alphabet songs, and a bit about the history of such things. It seems like these songs must be used primarily to teach the lexicographic ordering of the letters; I suppose the Latin alphabet is ordered in keeping with the Greek alphabet, and so forth, but why did the early alphabets get placed in the order that they did? Saying “numeric value”just begs the question (after all, then why those values?).

It also seems a bit odd that Twinkle Twinkle Little Star is song for the alphabet. It also seems like the alphabet song should be related to the zed/zee distinction.

And not too surprisingly, Wikipedia has an article about the Alphabet Songsong). Wikipedia knows too much (although they are still missing an article about superrigidity!).

Today in Geometry/Topology seminar, quasihomomorphisms $\Z \to \Z$ were discussed, i.e., the set of maps $f : \Z \to \Z$ such that $| f(a+b) - f(a) - f(b) |$ is uniformly bounded, modulo the relation of being a bounded distance apart. These come up when defining rotation and translation numbers, for instance.

Anyway, Uri Bader mentioned that these quasihomomorphisms form a field, isomorphic to $\R$, under pointwise addition and composition. I hadn’t realized that this is a general construction. Given a finitely generated group (with fixed generating set, so we have the word metric $d$ on the group), I can define a quasihomomorphism $f : G \to G$ by demanding $d(f(ab),f(a)f(b))$ be uniformly bounded, and where two quasihomomorphisms $f, g$ are equivalent if $d(f(a),g(a))$ is uniformly bounded. Let’s call the resulting object $\hat{G}$ for now.

What can be said about $\hat{G}$? For instance, what is $\hat{F_2}$?