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 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.

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 couple of seminar talks on

Lück, W.. Approximating $L\sp 2$-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 1994. 455–481. MR.

Here’s the main result in the paper. Let $X$ be a CW-complex, and filter $\Gamma = \pi_1 X$ as $\Gamma = \Gamma_1 \rhd \Gamma_2 \rhd \cdots$ with $[\Gamma_i : \Gamma_{i+1}] < \infty$ so that $\bigcap_i \Gamma_i = { 1 }$. Let $X_i$ be the cover of $X$ corresponding to the normal subgroup $\Gamma_i$.

Then, the limit of the “normalized” Betti numbers $\lim_{j \to \infty} b_j( X_i ) / [\Gamma : \Gamma_i]$ is equal to $b^{(2)}_j(X)$, the $L^2$ Betti number of $X$. In particular, the limit of the normalized Betti numbers is independent of the filtration! In other words, we have “approximated” the $L^2$ 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 on this subject,

Lück, Wolfgang. $L\sp 2$-invariants: theory and applications to geometry and $K$-theory. 2002. xvi+595. MR.

there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

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.

Jenn is a fabulous program for visualizing the Cayley graphs of finite Coxeter groups. The pictures are absolutely beautiful (oh, symmetry!).

I tried making bread, but with significantly less flour than neccessary (and therefore, far more water than needed). The result was very much like cooked paste. It was pointed out to me that since the essence of bread is flour, trying to get by with less flour was undermining the very essence of bread (and I find such arguments very satisfying).

I also made baklava again, and that turned out much better than the first time (which involved the baklava burning).

What can be said about the history of static electricity? Did Greek science know about it? Any medieval experiments with static electricity?

It’s sort of interesting that people knew about magnetism and electricity for hundreds of years before finding many good uses for that knowledge (granted, compasses and potentially batteries for electroplating, but these things are trinkets in our modern world so dependent on electricity); in contrast, the span between radiation and harnessing nuclear power was much shorter (although maybe our modern uses of nuclear power will seem like mere trinkets compared to the awesome uses to come). I guess this isn’t surprising—eh, nothing I say is surprising!

And after listening to Sufjan Stevens’ “A Good Man is Hard to Find,” I read the short story with the same title. I find myself liking “Seven Swans” more and more, and the short story by Flannery O’Connor was quite interesting. The short story of the crane wife (which is used to good effect on the Decemberists new album of the same name) is quite beautiful, too.

And last night, while doing some mathematics, I was also listening to an audiobook (well, podcast) rendition of Plato’s Republic; I had forgotten the thing about the ring that turned people invisible! It’s funny enough that this gets picked up in the Lord of the Rings, but just the idea of such a ring is so provocative—where did the idea come from?

And earlier this week, I was reading about king David’s “mighty men” and about the beautiful Abishag. I find it amusing how the names of these people (e.g., Glaucon in The Republic or Abishag) get remembered, with fame far beyond their expectation, I’m sure.

Cartan proved that every finite-dimensional real Lie algebra $\germ g$ comes from a connected, simply-connected Lie group $G$. I hadn’t known the proof of this result (and apparently it is rather uglier than one might hope), but

Gorbatsevich, V. V.. Construction of a simply connected group with a given Lie algebra. Uspekhi Mat. Nauk 1986. 177–178. MR.

gives a short proof of it, which I presented to the undergraduates in my Lie group seminar. I’ll sketch the proof now.

Theorem. For every Lie algebra $\mathfrak{g}$, there is a simply-connected, connected Lie group $G$ having $\mathfrak{g}$ as its Lie algebra.

First, if $\mathfrak{g} \subset \mathfrak{gl}(V)$, then the exponential map gives $U = \exp \mathfrak{g}$, and we define $G = \bigcup_{k=1}^\infty U^k \subset GL(V)$. It turns out $G$ is a Lie group, and $\mathfrak{g}$ is its Lie algebra.

If $\mathfrak{g}$ has no center, then $\rm{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$ is injective, so we have realized $\mathfrak{g}$ as a Lie subalgebra of endomorphisms of a vector space, and by the above, there is a Lie group $G \subset GL(\mathfrak{g})$ with $\mathfrak{g}$ as its Lie algebra. Taking its universal cover $\tilde{G}$ proves the theorem in this case.

Now we induct on the dimension of the center $Z(\mathfrak{g})$. Let $Z \subset Z(\mathfrak{g})$ be a one-dimensional central subspace of $\mathfrak{g}$, and construct a short exact sequence $0 \to Z \to \mathfrak{g} \to \mathfrak{g}’ \to 0$. But this central extension of $\mathfrak{g}’$ by $Z = \R$ corresponds to a 2-cocycle $\omega \in H^2(\mathfrak{g}; \R)$.

Lemma. Let $D : H^2(G;\R) \to H^2(\mathfrak{g}; \R)$ be the map which differentiates a (smooth!) $2$-cocycle of the group cohomology of $G$. The map $D$ is injective.

Consequently, we can find $f \in H^2(G;\R)$ with $Df = \omega$. Since $\dim Z(\mathfrak{g}’) < \dim Z(\mathfrak{g})$, by induction there is a Lie group $G’$ having $\mathfrak{g}’$ as its Lie algebra. We build the central extension of $G’$ by $\R$ using the cocycle $f$, namely, $0 \to \R \to G \to G’ \to 0$, where $G \cong G’ \times \R$ and the operation is $(g_1, t_1) \cdot (g_2, t_2) = (g_1 g_2, t_1 + t_2 + f(g_1,g_2))$. Since $Df = \omega$, it turns out that the Lie algebra corresponding to $G$ is $\mathfrak{g}$. We finish the proof by taking the universal cover $\tilde{G}$.

In seminar today, Okun pointed out the following interesting observation; for any finitely generated group $G$, you can define its growth series $G(t) = \sum_{g \in G} t^{\ell(g)}$, where $\ell(g)$ is the length of the shortest word for $g$. The first observation is that $G(t)$ is often a rational function, in which case $G(1)$ makes sense. The second observation is that $G(1)$ is “often” equal to $\chi(G)$. This is an example of weighted $L^2$ cohomology.

Grigorchuk’s group (and generally any group with intermediate (i.e., subexponential but not polynomial) growth) does not have a rational growth function; the coefficients in a power series for a rational function grow either polynomially or exponentially. This observation appears in

Stoll, Michael. Rational and transcendental growth series for the higher Heisenberg groups. Invent. Math. 1996. 85–109. MR.

More significantly, this paper constructs groups which, being nilpotent, have polynomial growth, but nonetheless have generating sets for which that the corresponding growth series is not rational.