The n-Category Café

”.. interesting reading.. ” well, hmm..
Merry Christmas!!

I’m happiest when people talk about the stuff in This Week’s Finds. So, here are some questions to get a discussion started. I think I know the answers to some of these — but I’m sure I don’t know the answers to others.

1. I wrote that the Universe became transparent to visible light 380,000 years after the Big Bang, more or less everywhere at once – but “since light takes time to travel, you’d see a transparent sphere around you, expanding outwards at the speed of light, with reddish walls”. But what would that actually look like? How can you even see something moving away from you at the speed of light? Isn’t that a contradiction?
2. Nothing is perfect; how would things look different when the Universe became transparent given that the Universe was hotter in some patches than others? A detailed answer requires knowing how big the temperature inhomogeneities were. But since we can see them, we should be able to say in detail how things looked.
3. What are the coolest things that people know or guess about Population III stars? What are the experts trying to figure out now?
4. Why is John Kelly Jr.’s work so controversial among card players, sports bettors, investors, hedge fund managers, and economists? Hint: it’s not the formula I mentioned, but the Kelly criterion, that all the fuss is about.
5. What are the most interesting recent papers on Klein geometry — the ones my friend Rafe must have been alluding to?

David Murphy sent me the following email, which he has kindly allowed me to post:

Dear John

As an ex-category theorist I’ve always enjoyed your columns, and I’d like to take the opportunity to thank you for them.

As a current mathematican finance person, however, I’d like to raise a slight note of concern. It isn’t that what you say is wrong per se, it is just that you suggest that financial systems are somehow like the probabilistic ones we meet in, say, statistical mechanics. That isn’t quite so. The reason is that financial markets are created by people trading. People believe a wide diversity of things, including that this or that stock is going up or down, that the market is calm or in crisis, or, in particular, that this or that theory describes the markets behaviour. Thus there is a feedback loop between theory and behaviour that makes finance profoundly different to most ‘scientific’ situations.

One well known example of this is the success of the Black Scholes formula for option pricing: before B&S, warrant prices were all over the place. Now they fall neatly into the line ‘predicted’ by B&S since many option buyers trade using that formula or its descendants. But, and here is the cute part, Black, Scholes and their co-worker Merton lost money trading off their formula in the early days of its development.

Why? Because an arbitrage relationship only holds if it is actually being traded. Some skilled traders put a trade on then tell the world about it, so that other people’s actions will cause the desired market movement. The market isn’t an abstract object whose behaviour makes sense without consideration of the actions of market participants.

I hope this perspective is helpful.

Kind regards

David

Hello,

This is probably too off-topic and semi-laypersonish… yet I couldn’t resist after I spotted Ehresmann connection and Cartan rolling here.

(I’ve given up on math and German EliteBildung a decade ago, yet this magnificent place (+TWF) threatens to get me hooked again sooner or later…)

One reason why I dropped out is that I don’t have enough patience to study unnecessarily messy stuff like the tensor “calculus” (be it in the “physicist’s notation” or the even more bunk “mathematician’s notation”). I made my own notation to juggle Laplacians etc. and then got tired of it all.

So, here’s my big puzzlement:

Why does the standard textbook and the standard paper start with the (Koszul) covariant derivative on vectors and not covectors?
1) Since the differential of a function is covector, this should be the natural place to proceed with calculus.
2) Torsion, curvature, and the Levi-Civita-Koszul construction would exhibit no Lie bracket terms when doing it on covectors. (It is an artefact of the product rule when switching to vector fields.)

To illustrate what I’m talking about, here’s my TeX-free definition of torsion:

Notation: C the smooth functions, T’ the C-module of covector fields, d: C–>T’ differential, T’*T’ tensor product, A: T’*T’–>T’^T’ projector on antisymmetric tensors.

Let D: T’–>T’*T’ be a covariant derivative, i.e. “Flori’s total covariant derivative starting at the right end”.
Then it is easy to check that the map ADd: C–>T’^T’ is a derivation (ADd is “taking the antisymmetric part of the Hesse form”).
Hence, by the universal property of d, there exists a unique tensor morphism t:T’–>T’^T’ with ADd=td.

Definition: t is called the torsion tensor of D.

Exercise: Check that t is the dual of the torsion in the “mathematician’s notation”.

Chris Weed emailed me the following remarks, which has kindly allowed me to post:

A physical perspective on Cartan geometry occurred to me, and I wonder if it motivates some of the interest in Cartan’s formulation. The idea is this: Consider the “model space” to represent a putative equilibrium configuration — the “target” of some kind of relaxation process. One might then ask, especially with quantum gravity in mind, what would underlie a given configuration. If one thinks in terms of a substructure, along the lines of LQG or Volovik’s condensed matter models, what sort of substructure would prefer to relax to a deSitter spacetime in the classical limit?

In a later email, he added:

I haven’t been going to n-Category Café often enough. In particular, this recent post by David Corfield was mildly electrifying:

• David Corfield, Common applications.

The “second strain of explanation” he discusses has been a pet notion of mine for about 15 years, and was directly inspired by ruminations on the writings of Edwin Jaynes combined with consideration of general relativity and field theory. I read Ariel Caticha’s paper with considerable excitement shortly after it appeared on the arXiv.

It is interesting that this theme of Corfield’s post is not unrelated to the remark in my previous message. Actually this isn’t so surprising, because the theme lurks in the background of nearly all my thinking about physics these days. In my opinion it has strong implicit connections to the Machian outlook as explicated by Julian Barbour. Consider the problem of inference he sets forth in the powerfully evocative introduction to Volume 1 of his Absolute or Relative Motion. One is confronted with a collection of snapshots of allegedly successive particle configurations, with no a priori coordinating labels to indicate a correspondence between them, and one is asked to place the snapshots in order and in “appropriate” relative orientations. It almost goes without saying that this problem has a deeply Bayesian flavor.

Mach’s philosophy also has strong logical and historical connections to the ongoing interest in inductive inference, which in some people’s minds has been substantially rehabilitated by the Bayesian approach. I have always had a heavily Popperian outlook, and my sense has been that Jaynes himself regarded Bayesian inference as the central tool in effective learning by trial and error, or conjecture and refutation. My understanding is that for inductivists, the notion that we learn by trial and error alone has always seemed untenable, because there is no evident reason why it should ever be effective unless the data can in some sense “speak to us”. As Popper himself once observed, if all one’s trials end in failure, after a while one wonders where to turn, i.e., whether one has learned anything at all that can guide future trials. All this seems quite close to David’s current concerns, as exemplified in:

• David Corfield, Macintyrean rationality of traditions and machine learning.

See the following, which cites Caticha’s paper:

• Thomas Marlow, Relationalism vs. Bayesianism

Let’s hope that Derek’s paper marks a watershed in the movement to have rodents properly represented in mathematics. The explication of the 3 conditions of a Cartan connection on pp. 16-17 are wonderfully clear thanks to that hamster.

If we ever get back to categorifying Klein geometry, will there then be scope to put 2-hamsters inside weak quotients of 2-groups, without harming them in any way?

As a warm-up we might look at how a 2-plane might 2-roll on a lumpy bumpy 2-surface.