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.

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).

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

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

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.