## Forget Me Not. Undo.

My Forget Me Not plug-in for Safari was reviewed on MacWorld!

It’s great to read the comments and find out what users wish were different: a lot of people don’t just want to Unclose Window but also Unclose Tab.

For me, I really would have liked to have Unclose Window be underneath the Edit menu, but because the undo hierarchy is linked to the window (i.e., when you switch windows, the possible things you can undo changes) it isn’t possible to undo the very destruction of the window itself.

The whole idea of “undoing” something is very amusing, especially the modern sense; the old sense of “undoing” something by destroying it is quite dissimilar from the idea of restoring to a previous state.

## Anime. Cats. Sufjan Stevens, again.

I watched Neon Genesis Evangelion this summer again, and I’m watching El Hazard now. I am extremely tempted to purchase Mysterious Cities of Gold DVD’s—does anyone else remember how awesome that was?

My cat Tasha is beautiful, and I will miss her while I am in Berkeley.

I am extraordinarily excited by the possibility that Sufjan Stevens might, in his epic quest to author a musical tribute for all 50 states, choose Minnesota next. Perhaps I am swayed by the fact that I am listening to Sufjan Stevens’ The Avalanche, and realizing again that I rather love him.

And I spend quite a bit of time brushing my teeth everyday, but I rather rarely discuss toothbrushing technique, or, for instance, how I learned to brush my teeth.

## I’m in California. I read Digital Fortress.

I’ve made it to Los Altos: I’m going to be staying with my dad, but during the next couple weeks visiting Berkeley to go to a conference, and to meet up with my advisor.

I ended up talking Route 22 on VTATransit_Authority#Bus_routes), and then walking a few miles to go here, in the dark, using GPS to guide me. I was amazed that this method worked!

I guess I should warn you that I am about to reveal plot details of Dan Brown’s books.

I read Digital Fortress by Dan Brown, which was an unfortunate use of time. That book, for starters, is isomorphic to the Da Vinci Code; they are both about a female cryptologist, who gets involved with a university professor, who is himself dragged into a global conspiracy. Humorously, at one point, they use a 5-letter password (which just happens to the female cryptologist’s name, just like in the Da Vinci Code). And like every Dan Brown book, this book also happens to begin with someone dying, who, as the holder of a secret, tries to reveal his secret before he dies.

The more unfortunate thing was the portrayal of mathematics and computer science in Digital Fortress. Terrifying, really.

## Public transportation is awesome.

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.

## Non-arithmetic lattices.

Gromov, M. and Piatetski-Shapiro, I.. Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. 1988. 93–103. MR.

Vinberg, …

Margulis’ amazing arithmeticity theorem says that irreducible lattices in Lie groups of high ($>2$) rank are arithmetic. But ${\rm SO}(n,1)$ has rank 1, so a question is how to produce non-arithmetic lattices. For ${\rm SO}(3,1)$, there are non-arithmetic lattices coming from hyperbolic knot complements.

G–P-S 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 $T^2$’s in $S^4$ which are hyperbolic, and maybe these would provide some examples.

## I’m back home, in Chicago.

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.

## Euler characteristic of closed hyperbolic 4-manifolds.

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).

## Tasha drops things in water.

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.

## Kolmogorov complexity.

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.

## And I was taking a bath…

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.

## More car trouble.

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.

## Detecting “cat-like” typing.

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).