Archive for February, 2007

Blogging Bounty

It appears that the new group blog Noncommutative Geometry really is the blog of Alain Connes, although to be fair it was set up by Arup Pal, according to Never Ending Books. There must be some sort of conspiracy going on here, because Terence Tao also has a new blog. And what are these great mathematicians blogging about? You guessed it: physics.

On Connes’ blog, today’s post by Goss is about the Bost-Connes reinterpretation of the Riemann zeta function as a statistical mechanical partition function.

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Riemann Rumbling On

I’ve just finished reading the delightful book by Sabbagh, Dr Riemann’s zeros. Although not a mathematician himself, Sabbagh competently launches into equations and diagrams, which he clearly explains for the lay reader. He spent a lot of time interviewing experts in the Riemann Hypothesis, to the point of attending their lectures, and he recorded the conversations.

The interaction of physics and the Riemann Hypothesis started with a memorable event, recounted in the book. Freeman Dyson was having afternoon tea as usual one day at Princeton in 1972. He was introduced to the visitor, Montgomery, who had been looking at the average gap in a long list of $\zeta$ zeroes. Montgomery mentioned his formula for the pair correlation, namely

$1 – ( \frac{\textrm{sin} \pi u}{\pi u} )^2$

at which point Dyson exclaimed that this was precisely the density of the pair correlation of eigenvalues of random matrices in the GUE.

Sabbagh was impressed by the awe that many mathematicians had for the Riemann Hypothesis. Conrey explained that the Riemann hypothesis is the most basic connection between addition and multiplication that there is, and Connes said: it is a basic primitive question about the adelic line which we don’t understand. It is a question about the way addition is fitting with multiplication.

Reminds one of categorical distributive laws, heh? Recall that for us addition and multiplication are monads (please don’t tell me you’ve forgotten about those). Anyway, a morphism $+ \times \rightarrow \times +$ which describes the commutativity of monads is a distributive law (wow – wikipedia is on to it). These are entities we need to think about in the context of quantum topos theory, because weak distributivity is the thing that separates quantum logic from the intuitionistic logic of an ordinary topos.

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Riemann Rambling On

Amongst physicists, Riemann is best known for the concept of metric. But his one paper on number theory, where he defined the zeta function, was not the only work he did on the subject. Riemann actually computed some zeroes of $\zeta (s)$ himself. This was unknown for about 60 years, until Siegel went through some work of Riemann’s and found a key to computing zeroes simply.

The Z function is defined by

$Z (t) = e^{i \theta (t)} \zeta (\frac{1}{2} + it)$

where

$\theta (t) = \textrm{arg} (\Gamma (\frac{2 i t + 1}{4})) – \frac{\textrm{log} \pi}{2} t$

The real zeroes of $Z (t)$ are the zeroes of $\zeta (s)$ on the critical line $s = \frac{1}{2} + it$. Positive real values of $t$ for which the $\zeta$ function is real are known as Gram points. By looking for pairs of Gram points, one can narrow down an interval where a zero of $\zeta$ must lie.

It is still rather impressive that Riemann managed to compute zeroes this way, with pen and paper, needing numbers such as the square root of 2 to something like 30 decimal places.

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Motive Madness

There are two papers by A. B. Goncharov, namely

1. Multiple polylogarithms and mixed Tate motives
2. Periods and mixed motives,

which are referred to by Brown in his paper on multiple zeta values and period integrals, as an excellent study of the punctured sphere moduli M(0,4) $\simeq \mathbb{P}^1 \backslash \{ 0,1, \infty \}$ and its finite covers based on roots of unity.

Hmmm. Maybe this post should be called M Theory Lesson 18. These topics (motives, number theory, gluon amplitudes etc.) are getting awfully mixed up! No, never mind. I’d better get back to reading, I guess. I must be crazy. It’s a gorgeous day outside…

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M Theory Lesson 17

In the recent comments, Mahndisa wondered what I meant by mentioning Euler’s relation

$e^{i \pi} = -1$

in M Theory Lesson 16. To be honest, it isn’t entirely clear to me. But rather than going ahead and expanding the exponential in the usual way, I was asking why this function has the properties it does. We need to view Euler’s relation anew, to see complex analysis in the light of diagrammatic reasoning. Non-standard analysis may take us a long way into topos methods, but it doesn’t dissect the good old complex plane in the simple geometric way we require. Remember that those associahedra tilings were about real moduli. Now we need to understand the complex case, using 2-operads.

2-operads involve two-level trees. If we replace the upper level by the ordinal which counts the number of leaves, we obtain one-level trees with labels. It is known that such trees are appropriate for complex moduli. Now we would like to take the 2-operad polytopes, such as the hexagon, and tile spaces with them.

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Parrot Smuggling Alert

Would the anonymous North American who found this blog by googling ‘keas as pets’ please desist with such disrespect for these rare birds. Unfortunately, your kind are all too common. Illegal capture and trading of kea are one of four main causes for their decline over the last few decades. The other causes are habitat degradation, introduced mammal nest predation and human killings (shooting or poisoning).

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M Theory Lesson 16

Recall that Leinster’s Euler characteristic for 1-categories could take on rational values. But we need more numbers than that. How might numbers be extended into the complex plane? We suspect that at some point higher categories will be involved.

For a finite directed graph, $\chi$ is simply given by $V – E$. For example, for the five arrow graph
we have that $\chi = -1$. Observe that this is the simplest way to obtain an Euler characteristic of -1 using circuit free graphs. An extra arrow is needed to make the diagram a category. What would it mean to take a square root of this diagram? Numbers must be represented by diagrams, so we must ask this question. One guess would be to call a triangle, with a 2-arrow in the interior, the number $i$. Let’s think about it. Everybody remembers Euler’s relation

$e^{i \pi} = -1$

Thinking of the logarithm instead, can we turn the multiplication (composition) into addition?

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Luscious Langlands II

Since domains on the complex plane might really represent moduli, and we know everything should be about categories at the end of the day, it would be better to replace the eigenfunction $f$ with a more sheaf theoretic concept. As Kapustin and Witten say in their abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions.

Yes, N=4 SUSY Yang-Mills turned up when we were worrying about twistor string theory and calculating gluon amplitudes. Actually, these days the geometric Langlands conjecture is about an equivalence of categories, namely derived categories of sheaves. Schreiber has been blogging about such things. But where did the number theory go in all this String geometry? Isn’t that what this is really about?

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Hot Dog Stars

From Physics Web today we hear the news: magnetic heating could be playing a much more prominent role in the evolution of neutron stars than previously expected, claim astrophysicists in Spain and the US … This could cause astrophysicists to rethink current theories of how neutron stars cool.

José Pons and colleagues used both satellite X-ray and ground based radio telescopes to investigate magnetic heating, which appears to be occurring in neutron stars. Louise Riofrio will be pleased to hear it. Another prediction comes true!

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Luscious Langlands

There are very few papers that have remained in my possession, after years of travelling and moving about. One of these is Stephen Gelbart’s An Elementary Introduction to the Langlands Program (Bull. Amer. Math. Soc. 10, 2 (1984) 177-215). These days there is no need to canvas mathematicians with a dim hope that they will see some connection of this to physics. I still understand very little of the Langlands program, but no longer for want of articles on the subject. Actually, I met Langlands once, back in the mid 90s. I had no idea who he was, much to the shock of the mathematicians I was socialising with at the time. Really cool guy – looks 20 years younger than he is.

Anyway, Gelbart’s paper begins with a quote by Browder: that the possible number fields of degree n are restricted by the irreducible infinite dimensional representations of GL(n) was the visionary conjecture of R. P. Langlands. The basic concept needed is that of an L-function associated to a representation $\pi$. We can start with the Artin version, which depends on a morphism $\sigma: G \rightarrow GL_{n}(C)$ for a certain Galois group $G$, and takes the form

$L (s, \sigma) = \prod_p (\textrm{det} (I_n – \sigma (F_p) p^{-s}))^{-1}$

where $I_n$ is an identity and $F_p$ is a Frobenius map. Oh yes, this is supposed to look a bit like the Riemann zeta function. Mahndisa will be pleased to know that the p-adics soon show up, what with primes floating about all over the place.

Langlands set about finding a correspondence $\sigma \rightarrow \pi (\sigma)$. The $n = 2$ case ends up being about automorphic forms. Remember that pretty fundamental domain, related to the modular group, that came up when we were talking about the moduli space for the punctured torus? One considers nice functions

$f(z) = \sum_{1}^{\infty} a_n e^{2 \pi i n z}$

with respect to such domains, so that when the set of $a_n$ is multiplicative, $f$ is an eigenfunction for Hecke operators.

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