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Category: stories

New world record obscurification

Published April 7, 2008 by lievenlb

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?

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the secret revealed…

Published March 26, 2008 by lievenlb

Often, one can appreciate the answer to a problem only after having spend some time trying to solve it, and having failed … pathetically.

When someone with a track-record of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”, I’m happy to spend the remains of the day trying to make sense of the apparent nonsense

Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

I expected the j-coefficients modulo 70 (or their squares, or their partial sums of squares) to reveal some hidden pattern, like containing the coefficients of Leech vectors or E(8)-roots, or whatever… and spend a day trying things out. But, all I got was noise… I left it there for a week or so, rechecked everything and… gave up

Subject:   Re: mystery message
From:  lieven.lebruyn@ua.ac.be
Date:  Fri 21 Mar 2008 12:37:47 GMT+01:00
To:    mckayj@Math.Princeton.EDU
    
i forced myself to recheck the calculations i did once after receiving your mail.
here are the partial sums of squares of j-coefficients modulo 70 for the first 
100 of them

[0, 46, 26, 16, 32, 62, 38, 3, 53, 13, 63, 39, 29, 59, 45, 10, 60, 40, 30,
 10, 40, 26, 6, 56, 42, 22, 68, 48, 48, 64, 64, 45, 25, 15, 31, 31, 67,
 47, 7, 21, 51, 31, 31, 61, 21, 1, 17, 12, 2, 16, 46, 60, 20, 10, 54, 49,
 63, 63, 53, 29, 29, 23, 13, 13, 27, 27, 17, 7, 67, 43, 43, 52, 42, 42,
 16, 6, 42, 42, 42, 36, 66, 32, 62, 52, 66, 66, 0, 25, 5, 5, 35, 21, 11,
 11, 57, 57, 61, 41, 41]

term 24 is 42...
i still fail to see the significance of it all.
atb :: lieven.

A couple of hours later I received his reply and simply couldn’t stop laughing…

From:  mckay@encs.concordia.ca
Subject:   Re: mystery message
Date:  Sat 22 Mar 2008 02:33:19 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

I apologize for wasting your time. It is a joke
depending, it seems, on one's cultural background.

See the google entry:

Answer to Life, the Universe, and Everything

Best, John McKay

Still confused? Well, do it!

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Monstrous Easter Egg Race

Published March 23, 2008 by lievenlb

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself…

From:  mckayj@Math.Princeton.EDU
Date:  Mon 10 Mar 2008 07:51:16 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

The secret of Monstrous Moonshine and the universe. 


Let  j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion 
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

Background: w_25 of page x of the preface of Conway/Sloane book SPLAG 

Also in Chapter 27:
The automorphism group of the 26-dimensional Lorentzian lattice
The Weyl vector w_25 of section 2.

Jm

I realize that all of you will feel frustrated by the fact that most university libraries are closed today and possibly tomorrow, hence some help with the background material.

SPLAG of course refers to the cult-book Sphere Packings, Lattices and Groups.

26-dimensional Lorentzian space $\mathbb{R}^{25,1} $ is 26-dimensional real space equipped with the norm-map

$|| \vec{v} || = \sum_{i=1}^{25} v_i^2 – v_{26}^2 $

The Weyl vector $\vec{w}_{25} $ is the norm-zero vector in $\mathbb{R}^{25,1} $

$\vec{w}_{25} = (0,1,2,3,4,\ldots,22,23,24,70) $ (use the numerical fact that $1^2+2^2+3^2+\ldots+24^2=70^2 $)

The relevance of this special vector is that it gives a one-line description for one of the most mysterious objects around, the 24-dimensional Leech Lattice $L_{24} $. In fact

$L_{24} = \vec{w}^{\perp}/\vec{w} $ with $\vec{w}^{\perp} = { \vec{x} \in \Pi_{25,1}~:~\vec{x}.\vec{w}=0 } $

where $\Pi_{25,1} $ is the unique even unimodular lattice in $\mathbb{R}^{25,1} $. These facts amply demonstrate the moonshine nature of the numbers 24 and 70. Apart from this, the previous post may also be of use.

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