Sometimes I’m scared that, at some point in my past, I opened a pair of parentheses without closing them. Even worse, I’m sure I’ve feared this very thing in the past.

Then again, maybe this is the common fear of all schemers: that our whole lives might now be a parenthetical comment.

I gave a couple of seminar talks on [1].

Here’s the main result in the paper. Let be a CW-complex, and filter as with so that . Let be the cover of corresponding to the normal subgroup .

Then, the limit of the “normalized” Betti numbers is equal to , the Betti number of . In particular, the limit of the normalized Betti numbers is independent of the filtration! In other words, we have “approximated” the invariant by a limit of finite-dimensional approximations.

The awesome thing about this result is how “easy” the proof is; it’s just some linear algebra (eh, functional analysis), but I don’t claim to have a very conceptual understanding of why it is true. In the big book [2] on this subject, there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

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 [1].

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 Song. Wikipedia knows too much (although they are still missing an article about superrigidity!).

Today in Geometry/Topology seminar, quasihomomorphisms were discussed, i.e., the set of maps such that 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 , 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 on the group), I can define a quasihomomorphism by demanding be uniformly bounded, and where two quasihomomorphisms are equivalent if is uniformly bounded. Let’s call the resulting object for now.

What can be said about ? For instance, what is ?

There are some questions about outer space that I would like to be able to answer. Some nice survey articles look to be [1] and also [2].

Here is a ridiculously simple question I have wondered about: given , say with , how can I tell if and are conjugate? I suspect I’m being stupid here.

In light of my recent comments on LINCOS and communicating with extraterristrials, I found the article [3].

Putnam also makes use of the idea of mathematicians from other planets, to more philosophical ends.

I’m fond of the Pineapple Sauce Pancake graph: the vertices are English words, and there is an edge from to if is also an English word (e.g., “pan” and “cake” are English words, and there is an edge from “pan” to “cake” because “pancake” is also an English word).

To play around with this, I wrote a Javascript program, complete with a Web 2.0 logo–which reminds me, I wonder if there is an interpreter for the programming language logo, written in Javascript?

Anyway, what I really wanted to do was to make a wall-sized picture of the Pineapple Graph, but Graphviz isn’t quite able to handle it, but maybe with some tweaking, I’d be able to produce a beautiful poster.

I’ve spent a lot of time constructing languages (Kisonef and Naedari being my favorites); in a similar vein, I also tried to create a language that an alien civilization would be able to understand. I had hoped to put a message written in my universal language in a conspicuous place (say, on a college campus), just to test if what I made really was understandable, even to humans!

But I never got around to that, and plenty of other people have done exactly that. This is related to the following question: state and prove a theorem in such a way that an alien would be able to follow your proof.

But whoa! I found out that Freudenthal (the mathematician) did the same thing: he created LINCOS. Bizarre. I also enjoyed looking at this image that we sent into space and trying to imagine what the aliens must think of people who write with such strange characters.