The n-Category Café

thanks for the illuminating answer. If one looks at this webpage one sees two interesting facts: the Halley comet seems to spend quite a lot of time of its 76 years of revolution rather far away from the sun, so not much subject to its radiation. As far as I remember (I was fortunately old enough to watch tv last time Halley visited us!) the tail starts to exist when it arrives near the sun. The second observation is that its orbit seems to be tangent to earth’s orbit!. Do they run through their orbits in the same direction? Is there a way to know Halley’s age?

Maybe there exist more energetic particles created by the big splash having much longer periods of revolution, spending a lot of time in the outer space, hence keeping safely their water? do we know which is the biggest possible “radius” of a body in orbit around the sun?

Yael writes:

… the Halley comet seems to spend quite a lot of time of its 76 years of revolution rather far away from the sun, so not much subject to its radiation.

I’m pretty sure that’s typical of all comets that remain bright enough for us to see for a long time. Comets lose mass when they get near the sun, so I just don’t think there are comets that spend their whole time as close to the Sun as, say, the Earth does. I can imagine a comet getting knocked into such a small orbit, but then it would lose all its volatile material — the stuff that makes the ‘tail’ — fairly soon.

Its orbit seems to be tangent to earth’s orbit! Do they run through their orbits in the same direction?

No, according to Wikipedia the orbit of Halley’s comet is retrograde: it goes around the Sun the opposite way from Earth and all the planets.

Is there a way to know Halley’s age?

Sure, you look it up in Wikipedia.

Just kidding — this doesn’t actually work. But, it was worth a try. Here’s what they say:

Halley’s comet may have been recorded in China as early as 467 BC, but this is uncertain. The first certain observation dates from 240 BC, and subsequent appearances were recorded by Chinese, Babylonian, Persian, and other Mesopotamian texts.

Halley is classified as a short period comet (a descriptor for comets with orbits lasting 200 years or less). However, its orbit is such that it is believed to have been originally a long period comet whose orbit was perturbed by the gravity of the giant planets and sent into the inner Solar System. It gives its name to the ‘Halley group’ of comets, which share these orbital characteristics.

If Halley was once a long period comet, it is likely to have originated in the Oort Cloud, a sphere of cometary bodies which has its inner edge at 50,000 AU. This distinguishes it from most other short period comets, which originate instead from the Kuiper Belt, a flat disc of icy debris between 38 AU (Pluto’s orbit) and 50 AU from the Sun.

In 1989, Boris Chirikov and Vitaly Vecheslavov performed an analysis of 46 apparitions of Halley’s Comet taken from historical records and computer simulations. These studies showed that the comet’s dynamics follow a simple area-preserving map similar to the standard map. Its dynamics were shown to be chaotic and unpredictable on long timescales. Halley’s projected lifetime, as determined by differential escape, is roughly 10 million years.

So, I guess nobody knows how old it is, but on very hand-wavy grounds I’d suspect no more than 10 million years. That’s a very short time, as far as astronomy goes.

A famous example of a comet with a much shorter period than Halley’s comet is Comet Encke. It goes around the Sun every 3 years. I wonder how long it will last.

I don’t know any comet that stays closer to the Sun than Encke.

do we know which is the biggest possible “radius” of a body in orbit around the sun?

By radius, do you mean the radius of the body or the radius of the orbit? The second option is more interesting to me — in week222 I wrote about ‘Transneptunian objects’, which orbit the Sun at very huge distances. The most distant ones seems to live in the Oort cloud. This consists of lots of comets in orbit up to one light-year away from the Sun. That’s 50,000 times farther from Sun than the Earth is! Perturbations from nearby stars sometimes push these comets towards the Sun, and they become the comets we see. Then they lose their volatile materials and fizzle out, or get tossed back out.

Actually, I did try quite hard to think of a maths question, but I couldn’t come up with anything more intelligent than: dualities are very interesting, I wish I understood them better. And I didn’t wish to put off paying my respects pending a mathematical insight!

One thing I have observed: it seems to me that a lot of dualities can be classified broadly as Poincaré-type dualities, and a lot of others as Pontryagin-type dualities. In fact, most dualities seem to fall into one or the other class. But I don’t understand what relationship, if any, exists between these two classes.

On the plus side, at the prompting of David’s comment below, I just read and understood the Wikipedia article on Tannaka-Krein duality, so either the article or my brain has improved dramatically since I last read it.

The geology and astronomy are interesting, but I don’t have anything to say about them. I don’t think this is because I’m a hardcore mathematician; I have about the same dilettantish interest in geology and astronomy as in mathematics. In fact, it’s somewhat the other way round—I think your discussion of geology and astronomy would have to be more hardcore before I could think of questions. (This isn’t a criticism: this isn’t an astronomy blog, and I’d go elsewhere if I wanted hardcore astronomy. The current level is fine. And other people seem to have comments …. )

Just one other remark: the oceans $\rightarrow$ life thing. As far as I can tell, we have almost no idea what conditions give rise to life or how common they are are. (In fact, except at the most abstract level, I’d say we haven’t the remotest, foggiest notion, except that “not very common at all” seems to be the way the current evidence is pointing.) So whenever anybody suggests liquid water, or liquid generally, as a necessary condition (or, more bizarrely, but more frequently, a sufficient condition!) for life, I get kind of jumpy. Really discovering an independently evolved life would certainly be very interesting, but speculating about it in the absence of evidence … not so much. I always feel—What, liquid nitrogen geysers shooting material into orbit around Saturn aren’t amazing enough for you? This just comes up over and over again in every discussion of the planets, and it annoys me every time.

Tim wrote:

In fact, it’s somewhat the other way round—I think your discussion of geology and astronomy would have to be more hardcore before I could think of questions.

I hope this means the stuff I’m writing makes sense, so you don’t feel the need to ask ‘huh?’ type questions.

But there are also questions of another sort, where one reads something and it makes one start wondering… I find I can easily generate huge wads of fairly sensible questions of this sort about astronomy. They may be questions nobody can answer, but they’re not completely idiotic.

Like: “Which moons of other planets were probably formed by collisions? And if not that way, how did they form? Is the fact that our Moon is especially large compared to its planet related to the fact that it formed from a collision? And how much evidence is there for this collision theory, anyway? What are the other main theories?”

I could keep generating these for pages, and I don’t think it takes any vast knowledge of astronomy. It seems to require more specialized knowledge to generate equally promising questions about, say, Pontryagin duality. For subjects like that, it seems one has to start by fighting through several layers of ‘huh?’ type questions.

Just one other remark: the oceans → life thing. As far as I can tell, we have almost no idea what conditions give rise to life or how common they are are.

That’s true. But Carolyn Porco’s remarks on this subject are pretty sensible, given our overall stunning lack of knowledge of xenobiology. She writes:

But in all, it is almost unavoidable that liquid water is present somewhere below its surface. If so, we face the thrilling possibility that within this little moon is an environment where life, or at least its precursor steps, may be stirring. Everything life needs appears to be available: liquid water, the requisite chemical elements and excess energy. The best analogues for an Enceladan ecosystem are terrestrial subsurface volcanic strata where liquid water circulates around hot rocks, in the complete absence of sunlight and anything produced by sunlight. Here are found organisms that consume either hydrogen and carbon dioxide, creating methane, or hydrogen and sulfate — all powered not by the sun but by Earth’s own internal heat.

As far as I can tell, she’s not saying that water is necessary for life. She’s mainly saying that an Enceladus with a buried ocean of liquid water would be an environment a lot like some places on Earth that support life — especially since we already know it has organic chemicals. In particular, it seems more promising than Europa, because there’s more energy flowing through the system.

But, I agree that emphasizing the search for water and life in our Solar System is likely to be a recipe for disappointment, since it makes otherwise interesting places seem like ‘failed Earths’.

It’s a bit like the approach to the history of mathematics in China that concentrates on the question why didn’t the Chinese come up with the Greek concept of ‘proof’? You wind up ignoring the cool stuff that’s actually there.

But isn’t this charm (of Pontryagin duality) bought at the expense of decategorification? If a group is a category, shouldn’t we expect an equivalence with any dual, rather than an isomorphism?

And if abelian groups are really 2-categories of a kind…

Good point, David. Does the charm of Pontryagin duality really come at the expense of decategorification?

Doplicher–Roberts duality tells us how to recover a compact group $G$ from its symmetric monoidal C*-category of continuous unitary representations $Rep(G)$. But what happens when this group is abelian?

The special thing about abelian groups is that all their irreducible representations are 1-dimensional. So, we don’t really need all of $Rep(G)$ to recover $G$; the 1-dimensional reps are enough.

The interesting thing is that if someone hands you $Rep(G)$ as an abstract symmetric monoidal category, you can pick out the 1-dimensional reps without knowing anything of the fact that objects of $Rep(G)$ are secretly vector spaces! The 1-dimensional objects are the invertible objects: the objects $X$ that have objects $Y$ with

$$X \otimes Y \cong 1$$

where $1$ is the unit for the tensor product (namely, the trivial rep of $G$).

Let’s call the full subcategory of invertible objects $Rep(G)^\times$. It’s easy to see that the tensor product of invertible objects is again invertible, so $Rep(G)^\times$ is again a monoidal category — but one of a special sort, since every object has an inverse! Ulbrich and Laplaza call such a thing a category with group structure.

$Rep(G)^\times$ not a 2-group, since not every morphism is invertible. But that’s easily fixed, if we want to fix it: just throw out all the noninvertible morphisms! We get a 2-group — and indeed a symmetric 2-group: that is, a symmetric monoidal category where all objects and all morphisms are invertible.

Now, what you’re complaining about, quite rightly, is that I’m taking this symmetric 2-group and decategorifying it to obtain an abelian group, which is the Pontryagin dual $G^*$ of our compact abelian group $G$.

Somehow this last step — the decategorification — is not losing any essential information, since we’re still able to recover $G$ from $G^*$.

How does it work? Rather roughly, it must go like this. We start with a continuous 1-dimensional unitary representation of $G$ and we say “Oh, this is isomorphic to one where $G$ is acting on our favorite 1d Hilbert space, namely $\mathbb{C}$. So, it might as well be a continuous homomorphism $\rho: G \to \mathrm{U}(1)$. But that’s an element of $G^*$.”

The question is why we don’t lose any interesting information when we engage in this decategorification. Some symmetric 2-groups do have information not contained in their decategorifications! In general a symmetric 2-group is classified by:

• an abelian group $A$ (the group of isomorphism classes of objects),
• an abelian group $B$ (the group of automorphisms of any object),
• a stable 3-cocycle: an element of $H^5(K(A,3),B)$ which records the associator and braiding.

I think that in the case at hand we have $A = G^*$, $B = \mathbb{C}^\times$, and the trivial stable 3-cocycle.

Why the trivial one? Well, I can’t see what else it would be, and still have no information lost when we decategorify and get $G^*$.

I’d love to see you try your hand at an explanation of the Hanbury-Brown and Twiss effect!

I haven’t thought about this effect for a long time. I see the Wikipedia article pushes the classical explanation as being more intuitive than the quantum one; that’s a nice slant, but I’ve only even thought about the quantum explanation, so I find that more intuitive than the classical one! The quantum explanation also handles fermions, like electrons.

At the bottom of this all is the simple fact: identical bosons like to ‘bunch’, identical fermions like to ‘antibunch’.

Maybe you know how that works, but I’ll just say it again. There are really three cases: boltzons, bosons, and fermions. “Boltzons” — one of Jim Dolan’s charming terms — satisfy Maxwell–Boltzmann statistics, rather than Bose–Einstein or Fermi–Dirac. Boltzons are actually the most intuitive of the three cases, since this case happens a lot in everyday life.

Suppose you have two identical balls and two urns, and you randomly put each ball in an urn, with 50-50 odds. What happens? More precisely: what’s the maximum entropy state of two balls in two urns?

Someone guess the answer in in each of the three cases: boltzons, bosons and fermions! Boltzons are the ‘intuitive’ case.

No fair — you posted your reply while I was writing up mine, below.

Yeah, basic quantum mechanics. I’m pretty embarrassed by the questions now.

For the $p$-adics, I think we can again say that the fourth iteration of the Fourier transform brings us back to the identity, so I think again there are only four eigenvalues given by the fourth roots of unity, and it probably works out pretty much the same way as for the reals. I don’t know the $p$-adic analogues of the $\psi_n$ you wrote down, but I think it ought to be a simple calculation (but it could still be fun!).

The small epiphany I mentioned at the start of all this was seeing that these “Gaussians” for the reals and $p$-adics were closely connected with the Euler factors for the zeta function. You often see the functional equation for the zeta function written in the form

$$Z(s) = Z(1-s)$$

where

$$Z(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)$$

Let’s write, according to the Euler product identity,

$$Z(s) = \pi^{-s/2}\Gamma(s/2)\prod_p (1 - p^{-s})^{-1}$$

About a year ago I read that these factors $(1 - p^{-s})^{-1}$ were called “Euler factors” at the places $p$, and that that funny fudge factor

$$\pi^{-s/2}\Gamma(s/2)$$

should be regarded as the Euler factor at the archimedean place. That’s sort of made me go, “huh?” – didn’t see where that was coming from. But it occurred to me [this is the “epiphany”] that the fudge factor is actually

$$\int_{\mathbb{R}} e^{-\pi x^2} |x|^s d x$$

and that the other Euler factors arose also in this way, by multiplying the analogous “Gaussian” at the non-archimedean place by a multiplicative homomorphism and integrating. So that gave me the happy feeling that I have a little toehold on getting at the functional equation (not just for the classical zeta, but fancier things like $L$-functions attached to characters) by means of Fourier analysis on local completions. And I was hoping that these “Gaussians” had particularly pleasant characterizations which would make everything feel very forced and canonical (and that’s what motivated the stupid questions in the first place). Hm… could we see the $p$-adic Gaussian [that is, the characteristic function of the $p$-adic integers] as the “vacuum state” in some sense??

The deeper mathematics underlying the small epiphany I had is without question very well understood, but it’s always pleasant to notice such little factoids on one’s own.

Sorry, I meant

$$\pi^{-s/2}\Gamma(s/2) = \int_{\mathbb{R}} e^{-\pi x^2} |x|^s \frac{d x}{x}$$

and even then I don’t guarantee that an arithmetic mistake hasn’t crept in. :-P

John said:

‘functions that are even and ????’

Erm, even and zero? According to my sums $q\psi_0=x\psi_0$ and $p\psi_0=i x\psi_0$ so $(q+i p)\psi_0 =0$.

So, Todd: it sounds like you’re saying the ‘improved zeta function’

$$Z(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)$$

is a product over completions of $\mathbb{Q}$ of factors that are roughly Mellin transforms of Gaussians — or $p$-adic analogues thereof.

That sounds cool! But do you — or anyone reading this — know the conceptual importance of taking a Mellin transform of a Gaussian?

Up to annoying fudge factors, the Mellin transform is basically a Fourier transform not for the locally compact abelian group $(\mathbb{R}, +)$ but its multiplicative version $(\mathbb{R}^+, \times)$.

(It’s actually more like a Laplace transform, but I count that among the ‘annoying fudge factors’.)

On the other hand, the Gaussian plays a fundamental role in the Fourier theory of $(\mathbb{R}, +)$. You were overoptimistic in guessing that it was the only 1-eigenvector of the Fourier transform, but something very similar is true. $L^2(\mathbb{R}^n)$ comes with a god-given projective representation of the symplectic group $Sp(n)$, which contains the unitary group $\mathrm{U}(n)$, and the Gaussian is, up to a constant, the unique vector that’s fixed up to phase by all of $\mathrm{U}(n)$. This is a mathematical way of saying it’s the ‘ground state’ of the $n$-variable harmonic oscillator.

I’m sure someone understands all this stuff and how it’s related to number theory. For example, I know André Weil was interested in the projective representation of the symplectic group on $L^2(\mathbb{R}^n)$ — the so-called ‘metaplectic representation’. So, now I’ll wildly guess that he was also interested in the $p$-adic and adelic analogues of this thing.

But, I’d like to hear a simple story about what’s going on.

Of course, what I’m calling $Sp(1)$ is none other than $SL(2)$, a group greatly beloved by number theorists. This has got to be part of the story.