Last night, there was a terrible thunderstorm in Chicago; I’ve never seen so many trees on the road! I was supposed to land at Midway at 7:30pm last night, but we were diverted to Indianapolis, so I didn’t land in Chicago until 2:00am, and then I waited until 3:00am to get a taxi, so I didn’t get home until almost 4:00am. Crazy!

My house lost power last night, and today some places are still without power. In particular, Co-Op Market’s 53th street store was closed, merely displaying a sign “Closed No Power Mgmt.” Considering that their 47th street store shut down, and that their 55th still (?) lacks price scanners, I can only expect that this power outage is the final blow to Co-Op.

In contrast, the also-powerless-but-superior Hyde Park Produce Market was using a generator to power their cash registers (and to provide one very bright light in an otherwise dark store).

My cell phone still works, even after being dropped into water.

There is a program called PawSense for Windows which detects “cat-like” typing, and then prevents further keyboard entry.

I found some code for filtering keyboard events on Mac OS X, and I wanted to implement something similar. But this raises an interesting question: just what characterizes “cat-like” typing?

The PawSense website suggested that cat paws are very broad, and usually strike nearby keys simultaneously. Another idea is to detect “human-like” typing and then freeze the keyboard whenever non-human typing is detected (which has the useful feature of detecting a future version of cat with smaller paws).

This morning I went to the car to see if I could start it, and at least move it back and forth a bit (as I still don’t have my license). Fortunately, the car started! Unfortunately, the clutch doesn’t seem to do anything.

I am holding down the clutch while the car starts, but then I can’t shift into reverse: all I hear is gear-grinding. I can’t shift into first at all; the knob won’t even move there. With the car off, I shifted into reverse, and then started the car (okay okay, I realize now this was a truly stupid idea), but it merely lurched backward before the engine died. It’s just as if I let up on the clutch too quickly without enough gas…

I guess the clutch isn’t doing anythnig at all.

This must be a consequence of my having not driven it while I was away; the emergency brake is sort of loose feeling, the door locks are sticking, and, rather tellingly, the clutch sort of squeaks when I move it. I guess this means I will have it towed away again to be repaired again; at least the people at the transmission shop are very nice.

I really just want to learn how to drive. Someday, someday.

Two interesting things about taking a bath…

The first was, while singing in the bathtub, I hit a resonant frequency, and I wondered: what can be deduced about the shape of my bathtub (well, bathroom) from this frequency?

The second was that I heard something fall into Tasha’s water dish; thinking nothing of it, I was rather shocked (well, not literally shocked, but…) to find that it was my cell phone that had fallen into Tasha’s water. Uh oh.

And now I am trying to dry it off with a hairdryer.

Here are some very ill-thought-out ideas on Kolmogorov complexity.

We define a metric on the space of bit-strings . For a universal Turing machine , let be the "length" of the shortest program that outputs on input , or outputs on input . This should measure how difficult it is to "relate" and .

The ends of the metric space should correspond to infinite random bitstrings, and because choosing a different univeral Turing machine just replaces this metric space with one quasi-isometric to it, the ends should be preserved, so there will still be a lot of infinite random bitstrings

But obviously I haven’t thought about any of this very carefully: for instance, the triangle inequality probably only holds coarsely, because it depends on being able to concatenate programs.

Here’s a similar question. Usually, we start with a partial function which tells how to translate descriptions into objects; Kolmogorov complexity is then defined as . Any universal Turing machine gives a measure of complexity with the same asymptotics, i.e., and differ by a constant that depends only on and . Suppose I have another function so that has the same asymptotics: what more can be said about ?

There’s a stupid rigidity for computable functions (a computable function is still computable if its value is changed at finitely many places), and maybe these sort of questions could lead to a rigidity theorem for computability, a local Church-Turing thesis.

And having written this, I’m terrified at how similar I sound to Archimedes Plutonium. Now I’ll go to learn more about localization of spaces in the algebraic topology proseminar.

Often, Tasha picks something up (say, a pen, or a lego), carries it around, and then drops it into her water bowl. I have no idea what she is thinking when she does this. On the topic of cat thoughts, the Wikipedia article on cats observes:

Some theories suggest that cats see their owners gone for long times of the day and assume they are out hunting, as they always have plenty of food available.

I desperately hope that Tasha believes that I am out hunting (mathematics?). In any case, seeing her carry the legos around answers an old question of mine: about two months ago, I noticed that lego pieces were “mysteriously” appearing in my shoes. The Wikipedia article goes on to note that:

It is thought that a cat presenting its owner with a dead animal thinks it’s ‘helping out’ by bringing home the kill.

In other news, the welcome dinner for GCF went spectacularly well; afterwards, we played Loaded Questions, and I learned that people associate tildes with me to a much stronger degree than I would have believed.

By the Gauss-Bonnet theorem, the volume of a hyperbolic 4-manifold is proportional to its Euler characteristic. There are examples, constructed explicitly in [1] of hyperbolic 4-manifolds with every positive integer as their Euler characteristic. These examples are non-compact (with five or six cusps, I believe). But [2] observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in [3] that the Pontrjagin numbers of a hyperbolic manifold vanish. But the signature is a rational linear combination of those Pontrjagin numbers, so . And by Poincare duality, , so is even. A natural question to ask is: does there exist a hyperbolic 4-manifold with ? Now if such an also had , we would know the volume spectrum of closed hyperbolic 4-manifolds.

This certainly seems to parallel the case for 2-manifolds: all negative integers are the Euler characteristic of a hyperbolic 2-manifold, and all even negative integers are the Euler characteristic of a closed hyperbolic 2-manifold.

The vanishing of Pontrjagin numbers for hyperbolic manifolds also holds for pinched negative curvature under some conditions [1].

It is also a fact that the Stiefel-Whitney numbers vanish for a closed hyperbolic manifold (and the vanishing of the top Stiefel-Whitney class is the same thing as having even Euler characteristic).

I left California far too quickly: some people that I had really wanted to see I didn’t get to see. But I got to spend a lot of time with my dad, which was excellent, and the conferences and Berkeley itself were a lot of fun.

I understand why clutching functions are called clutching functions: a automobile’s clutch transmits rotation from one object to another under the control of the driver, and a clutching function likewise glues together two different rotations under the control of the mathematician.

Having been gone for two weeks, the beautiful cat Tasha has decided that my chair is her chair.

I saw a poster that described a play as “crunchingly witty.” This seems like a very strange sort of wittiness to me.

Margulis’ amazing arithmeticity theorem says that irreducible lattices in Lie groups of high () rank are arithmetic. But has rank 1, so a question is how to produce non-arithmetic lattices. For , there are non-arithmetic lattices coming from hyperbolic knot complements.

G–P-S [1] produces higher dimensional examples by taking two hyperbolic (arithmetic) manifolds, cutting along totally geodesic hypersurfaces, and gluing. Are there are examples of non-arithmetic hyperbolic manifolds without any totally geodesics hypersurfaces?

There are complements of ’s in which are hyperbolic, and maybe these would provide some examples.

I am still in California, and very much enjoying public transportation. Yesterday, I took AC Transit’s 65 bus (the “Euclid” bus) along an extremely (and therefore ironically) curvy road to get off the mountain of MSRI. (The “mountain of misery” belongs in a fantasy novel.)

There’s a lot of people in California I would still like to see.

Here is a really stupid question: If I have a wire (with a changing current) and I bend it around, and measure the induced current in (a finite number of) neighboring coils, can I determine anything about how I have bent the wire? Being more ridiculous, I will weave together wires to make fabric. How do the electrical properties of the fabric relate to its shape? That is, if I run current along one wire, and measure the induced current in other wires, can I deduce anything about how I have bent the fabric?

It strikes me as amusing to measure the speed of something by, say, attaching a magnet to the wheel and then seeing how quickly the magnet is moving past a coil. I wonder how sensitive this would have to be, say, to work as a bicycle speedometer.

Anyway, this is the stuff that bothers me on the bus. Right now, it is my inability to produce more examples of hyperbolic n-manifolds that is bothering me.