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New world record obscurification

Monday, April 7th, 2008

I’ve always thought of Alain Connes as the unchallengeable world-champion opaque mathematical writing, but then again, I was proven wrong.

Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros - Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus.

Here’s a generic half-page from a total of 558 pages (or rather 314, as the remainder consists of appendices, bibliography and indices…). I couldn’t find a single precise, well-defined and proven statement in the entire book.

4.2. Fractal Membranes and the Second Quantization of Fractal Strings
“The first quantization is a mystery while the second quantization is a functor” Edward Nelson (quoted in [Con6,p.515])

We briefly discuss here joint work in preparation with Ryszard Nest [LapNe1]. This work was referred to several times in Chapter 3, and, as we pointed out there, it provides mathematically rigorous construction of fractal membranes (as well as of self-similar membranes), in the spirit of noncommutative geometry and quantum field theory (as well as of string theory). It also enables us to show that the expected properties of fractal (or self-similar) membranes, derived in our semi-heuristic model presented in Sections 3.2 and 3.2. are actually satisfied by the rigorous model in [LapNe1]. In particular, there is a surprisingly good agreement between the author’s original intuition on fractal (or self-similar) membrane, conceived as an (adelic) Riemann surface with infinite genus or as an (adelic) infinite dimensional torus, and properties of the noncommutative geometric model in [LapNe1]. In future joint work, we hope to go beyond [LapNe1] and to give even more (noncommutative) geometric content to this analogy, possibly along the lines suggested in the next section (4.3).
We will merely outline some aspects of the construction, without supplying any technical details, instead referring the interested reader to the forthcoming paper [LapNe1] for a complete exposition of the construction and precise statements of results.

Can the AMS please explain to the interested person buying this book why (s)he will have to await a (possible) forthcoming paper to (hopefully) make some sense of this apparent nonsense?

Tags: Connes, geometry, noncommutative, Riemann
Posted in geometry | 2 Comments »

KMS, Gibbs & zeta function

Thursday, February 21st, 2008

Noncommutative geometry and the Riemann zeta function

Time to wrap up this series on the Bost-Connes algebra. Here’s what we have learned so far : the convolution product on double cosets

$\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} \backslash \begin{bmatrix} 1 & \mathbb{Q} \\ 0 & \mathbb{Q}_{> 0} \end{bmatrix} / \begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix}$

is a noncommutative algebra, the Bost-Connes Hecke algebra $\mathcal{H}$ , which is a bi-chrystalline graded algebra (somewhat weaker than ’strongly graded’) with part of degree one the group-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ . Further, $\mathcal{H}$ has a natural one-parameter family of algebra automorphisms $\sigma_t$ defined by $\sigma_t(X_n) = n^{it}X_n$ and $\sigma_t(Y_{\lambda})=Y_{\lambda}$ .

For any algebra together with a one-parameter family of automorphisms $\sigma_t$ one is interested in KMS-states or Kubo-Martin-Schwinger states with parameter $\beta$ , $KMS_{\beta}$ (this parameter is often called the ‘invers temperature’ of the system) as these are suitable equilibria states. Recall that a state is a special linear functional $\phi$ on (in particular it must have norm one) and it belongs to $KMS_{\beta}$ if the following commutation relation holds for all elements $a,b \in A$

$\phi(a \sigma_{i\beta}(b)) = \phi(b a)$

Let us work out the special case when is the matrix-algebra $M_n(\mathbb{C})$ . To begin, all algebra-automorphisms are inner in this case, so any one-parameter family of automorphisms is of the form

$\sigma_t(a) = e^{itH} a e^{-itH}$

where $e^{itH}$ is the matrix-exponential of the nxn matrix . For any parameter $\beta$ we claim that the linear functional

$\phi(a) = \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H})$

is a KMS-state.Indeed, we have for all matrices $a,b \in M_n(\mathbb{C})$ that

$\phi(a \sigma_{i \beta}(b)) = \frac{1}{tr(e^{-\beta H})} tr(a e^{- \beta H} b e^{\beta H} e^{- \beta H})$

$= \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H} b) = \frac{1}{tr(e^{-\betaH})} tr(ba e^{-\beta H}) = \phi(ba)$

(the next to last equality follows from cyclic-invariance of the trace map). These states are usually called Gibbs states and the normalization factor $\frac{1}{tr(e^{-\beta H})}$ (needed because a state must have norm one) is called the partition function of the system. Gibbs states have arisen from the study of ideal gases and the place to read up on all of this are the first two chapters of the second volume of “Operator algebras and quantum statistical mechanics” by Ola Bratelli and Derek Robinson.

This gives us a method to construct KMS-states for an arbitrary algebra with one-parameter automorphisms $\sigma_t$ : take a simple n-dimensional representation $\pi~:~A \mapsto M_n(\mathbb{C})$ , find the matrix determining the image of the automorphisms $\pi(\sigma_t)$ and take the Gibbs states as defined before.

Let us return now to the Bost-Connes algebra $\mathcal{H}$ . We don’t know any finite dimensional simple representations of $\mathcal{H}$ but, sure enough, have plenty of graded simple representations. By the usual strongly-graded-yoga they should correspond to simple finite dimensional representations of the part of degree one $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ (all of them being one-dimensional and corresponding to characters of $\mathbb{Q}/\mathbb{Z}$ ).

Hence, for any $u \in \mathcal{G} = \prod_p \hat{\mathbb{Z}}_p^{\ast}$ ( details) we have a graded simple $\mathcal{H}$ -representation $S_u = \oplus_{n \in \mathbb{N}_+} \mathbb{C} e_n$ with action defined by

$\begin{cases} \pi_u(X_n)(e_m) = e_{nm} \\ \pi_u(Y_{\lambda})(e_m) = e^{2\pi i n u . \lambda} e_m \end{cases}$

Here, $u.\lambda$ is computed using the ‘chinese-remainder-identification’ $\mathcal{A}/\mathcal{R} = \mathbb{Q}/\mathbb{Z}$ ( details).

Even when the representations S_u are not finite dimensional, we can mimic the above strategy : we should find a linear operator determining the images of the automorphisms $\pi_u(\sigma_t)$ . We claim that the operator is defined by H(e_n) = log(n) e_n for all $n \in \mathbb{N}_+$ . That is, we claim that for elements $a \in \mahcal{H}$ we have

$\pi_u(\sigma_t(a)) = e^{itH} \pi_u(a) e^{-itH}$

So let us compute the action of both sides on e_m when a=X_n . The left hand side gives $\pi_u(n^{it}X_n)(e_m) = n^{it} e_{mn}$ whereas the right-hand side becomes

$e^{itH}\pi_u(X_n) e^{-itH}(e_m) = e^{itH} \pi_u(X_n) m^{-it} e_m =$

$e^{itH} m^{-it} e_{mn} = (mn)^{it} m^{-it} e_{mn} = n^{it} e_{mn}$

proving the claim. For any parameter $\beta$ this then gives us a KMS-state for the Bost-Connes algebra by

$\phi_u(a) = \frac{1}{Tr(e^{-\beta H})} Tr(\pi_u(a) e^{-\beta H})$

Finally, let us calculate the normalization factor (or partition function) $\frac{1}{Tr(e^{-\beta H})}$ . Because $e^{-\beta H}(e_n) = n^{-\beta} e_n$ we have for that the trace

$Tr(e^{-\beta H}) = \sum_{n \in \mathbb{N}_+} \frac{1}{n^{\beta}} = \zeta(\beta)$

is equal to the Riemann zeta-value $\zeta(\beta)$ (at least when $\beta > 1$ ).

Summarizing, we started with the definition of the Bost-Connes algebra $\mathcal{H}$ , found a canonical one-parameter subgroup of algebra automorphisms $\sigma_t$ and computed that the natural equilibria states with respect to this ‘time evolution’ have as their partition function the Riemann zeta-function. Voila!

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Tags: Artin, Connes, Gibbs, noncommutative, partition function, representations, Riemann, states, zeta function
Posted in geometry | 2 Comments »

music of the primes (1)

Wednesday, February 13th, 2008

This semester, I’m running a 3rd year course on Marcus du Sautoy’s The music of the primes. The concept being that students may suggest topics, merely sketched in the book, and then we’ll go a little deeper into them.

I’ve been rather critical about the book before, but, rereading it last week (and knowing a bit better the limitations of bringing mathematics to the masses…) I think du Sautoy did a great job. Sure, it focusses too much on people and places and too little on mathematics, but that goes with the format.

I wanted to start off gently by playing the open-university dvd-series so that students would have a very rough outline of the book from the very start (as well as a mental image to some of the places mentioned, such as Bletchley Park, the IAS, Gottingen…). However, the vagueness of it all seemed to work on their nerves … in particular the trumpet scenes

Afterwards, they demanded that I should explain next week what on earth the zeroes of the Riemann zeta function had to do with counting primes and what all this nonsensical ‘music of the primes’ was about.

Well, here is the genuine music of the primes (taken from the Riemann page by Jeffrey Stopple whose excellent introductory text A Primer of Analytic Number Theory I’ll use to show them some concrete stuff (they have their first course on complex analysis also this semester, so I cannot go too deep into it).

Jeffrey writes “This sound is best listened to with headphones or external speakers. For maximum effect, play it LOUD.” But, what is the story behind it?

The Von Mangoldt function $\Lambda(n)$ assigns log(p) whenever n=p^k is a prime power and zero otherwise. One can then consider the function

$\Psi(x) = \frac{1}{2}(\sum_{n < x} \Lambda(n) + \sum_{n \leq x} \Lambda(n))$

which makes a jump at prime power values and the jump-size depends on the prime. Here is a graph of its small values

It’s not quite the function $\pi(x)$ (counting the number of primes smaller than x) but it sure contains enough information to obtain this provided we have a way of describing $\Psi(x)$ .

The Riemann zeta function (or rather $~(s-1)\zeta(s)$ ) has two product descriptions, the Hadamard product formula (running over all zeroes, both the trivial ones at -2n and those in the critical strip), which is valid for all complex s and the Euler product valid for all Re(s) > 1 . This will allow us to calculate in two different ways $\zeta'(s)/\zeta(s)$ which in turn allows us to have an explicit description of $\Psi(s)$ known as the Von Mangoldt formula

$\Psi(x) = x - \frac{1}{2}log(1 - \frac{1}{x^2}) - log(2 \pi) - \sum_{\rho} \frac{x^{\rho}}{\rho}$

where only the last term depends on the zeta-zeroes $\rho$ lying in the critical strip (and conjecturally all lying on the line $Re(x) = \frac{1}{2}$ . The first few terms (those independent of the zeroes) give a continuous approximation of $\Psi(x)$ but how on earth can we get from that approxamation (on the left) to the step-like function itself (on the right)?

We can group together zeta-zeroes $\rho=\beta + i \gamma$ with their comlex conjugate zeroes $\overline{\rho}$ and then one shows that the attribution to the Von Mangoldt formula is

$\frac{x^{\rho}}{\rho} + \frac{x^{\overline{\rho}}}{\overline{rho}} = \frac{2 x^{\beta}}{| \rho | }cos(\gamma log(x) - arctan(\gamma/\beta))$

Ignoring the term $x^{\beta}$ this is a peridodic function with amplitude $2/| \rho |$ (so getting smaller for larger and larger zeroes) and period $2\pi/ \gamma$ . If the Riemann hypothesis holds (meaning that $\beta=1/2$ for all zeroes) one can even split a term in this contribution of every zero as a sort of ‘universal amplitude’. What is left is then a sum of purely periodic functions which a physicist will view as a superposition of (sound) waves and that is the music played by the primes!

Below, a video of the influence of adding the first 100 zeroes to a better and better approximation of $\Psi(x)$ (again taken from the Riemann page by Jeffrey Stopple). Surely watching the video will convince anyone of the importance of the Riemann zeta-zeroes to the prime-counting problem..

Tags: apple, Riemann
Posted in general | 1 Comment »

vacation reading (2)

Tuesday, February 5th, 2008

math-books

Vacation is always a good time to catch up on some reading. Besides, there’s very little else you can do at night in a ski-resort… This year, I’ve taken along The Archimedes Codex: Revealing The Secrets Of The World’s Greatest Palimpsest by Reviel Netz and William Noel telling the story of the Archimedes Palimpsest.

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.
Essentially, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and “weighing” the stripes of the first figure against those of the second, evaluated in terms of a finite Egyptian fraction series. He considered this method as a useful heuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.
He was able to solve problems that would now be treated by integral calculus, which was formally invented in the 17th century by Isaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, see Archimedes’ use of infinitesimals.)
A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler’s Stereometria.
Some pages of the Method remained unused by the author of the Palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.
In Heiberg’s time, much attention was paid to Archimedes’ brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children’s puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Also I hope to finish the novel Interred with their bones by Jennifer Lee Carrell (though I prefer the Dutch title, “Het Shakespeare Geheim” that is, “The Shakespeare Secret”) on a lost play by Shakespeare, and have a re-read of The music of the primes as I’ll use this book for my course starting next week.

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Tags: Archimedes, Artin, books, geometry, mac, mathematics, puzzle, Riemann
Posted in general | 2 Comments »

un-doing the Grothendieck?

Wednesday, January 23rd, 2008

(via the Arcadian Functor) At the time of the doing the Perelman-post someone rightfully commented that “making a voluntary retreat from the math circuit to preserve one’s own well-being (either mental, physical, scientific …)” should rather be called doing the Grothendieck as he was the first to pull this stunt.

On Facebook a couple of people have created the group The Petition for Alexander Grothendieck to Return from Exile. As you need to sign-up to Facebook to use this link and some of you may not be willing to do so, let me copy the description.

Alexander Grothendieck was born in Berlin, Germany on March 28, 1928. He was one of the most important and enigmatic mathematicians of the 20th century. After a lengthy and very productive career, highlighted by the awarding of the Fields Medal and the Crafoord Prize (the latter of which he declined), Grothendieck disappeared into the French countryside and ceased all mathematical activity. Grothendieck has lived in self-imposed exile since 1991.
We recently spotted Grothendieck in the “Gentleman’s Choice” bar in Montreal, Quebec. He was actually a really cool guy, and we spoke with him for quite some time. After a couple of rounds (on us) we were able to convince him to return from exile, under one stipulation - we created a facebook petition with 1729 mathematician members!
If 1729 mathematicians join this group, then Alexander Grothendieck will return from exile!!

1729 being of course the taxicab-curve number. The group posts convincing photographic evidence (see above) for their claim, has already 201 members (the last one being me) and has this breaking news-flash

Last week Grothendieck, or “the ‘Dieck” as we affectionately refer to him, returned to Montreal for a short visit to explain some of the theories he has been working on over the past decade. In particular, he explained how he has generalised the theory of schemes even further, to the extent that the Riemann Hypothesis and a Unified Field Theory are both trivial consequences of his work.

You know what to do!

Tags: 1729, facebook, Grothendieck, mathematics, Perelman, petition, Riemann
Posted in general | 2 Comments »