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 $\Sigma^\star$. For a universal Turing machine $T$, let $d_T(x,y)$ be the “length” of the shortest program that outputs $y$ on input $x$, or outputs $x$ on input $y$. This should measure how difficult it is to “relate” $x$ and $y$.

The ends of the metric space $(\Sigma^\star, d_T)$ 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 $f : \Sigma^\star \to \Sigma^\star$ which tells how to translate descriptions into objects; Kolmogorov complexity is then defined as $C_f(x) = \min_{f(y) = x} |y|$. Any universal Turing machine gives a measure of complexity with the same asymptotics, i.e., $C_g(x)$ and $C_f(x)$ differ by a constant that depends only on $f$ and $g$. Suppose I have another function $h$ so that $C_h$ has the same asymptotics: what more can be said about $h$?

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

Ratcliffe, John G. and Tschantz, Steven T.. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math. 2000. 101–125. MR.

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

Ratcliffe, John G.. The geometry of hyperbolic manifolds of dimension at least 4. 2006. 269–286. MR.

observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in

Chern, Shiing-shen. On curvature and characteristic classes of a Riemann manifold. Abh. Math. Sem. Univ. Hamburg 1955. 117–126. MR.

that the Pontrjagin numbers of a hyperbolic manifold $M$ vanish. But the signature $\sigma(M)$ is a rational linear combination of those Pontrjagin numbers, so $\sigma(M) = 0$. And by Poincare duality, $\chi(M) \equiv \sigma(M) \pmod 2$, so $\chi(M)$ is even. A natural question to ask is: does there exist a hyperbolic 4-manifold $M$ with $\chi(M) = 2$? Now if such an $M$ also had $H_1(M) \neq 0$, 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:

Ratcliffe, John G. and Tschantz, Steven T.. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math. 2000. 101–125. MR.

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).