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Finding Moonshine

On friday, I did spot in my regular Antwerp-bookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me…

To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more, the more I convinced myself that it was a mere resampling of Symmetry and the Monster, The equation that couldn’t be solved, From Error-Correcting Codes through Sphere Packings to Simple Groups and the diary-columns du Sautoy wrote for a couple of UK-newspapers about his ‘life-as-a-mathematician’…

Still, I took the book home, made a pot of coffee and started reading the first chapter. And, sure enough, soon I had to read phrases like “The first team consisted of a ramshackle collection of mathematical mavericks. One of the most colourful was John Horton Conway, currently professor at the University of Princeton. His mathematical and personal charisma have given him almost cult status…” and “Conway, the Long John Silver of mathematics, decided that an account should be published of the lands that they had discovered on their voyage…” and so on, and so on, and so on.

The main problem I have with du Sautoy’s books is that their main topic is NOT mathematics, but rather the lives of mathematicians (colourlful described with childlike devotion) and the prestige of mathematical institutes (giving the impression that it is impossible to do mathematics of quality if one isn’t living in Princeton, Paris, Cambridge, Bonn or … Oxford). Less than a month ago, I reread his ‘Music of the Primes’ so all these phrases were still fresh in my memory, only on that occasion Alain Connes is playing Conway’s present role…

I was about to throw the book away, but first I wanted to read what other people thought about it. So, I found Timothy Gowers’ review, dated febraury 21st, in the Times Higher Education. The first paragraph below hints politely at the problems I had with Music of the Primes, but then, his conclusion was a surprise

The attitude of many professional mathematicians to the earlier book was ambivalent. Although they were pleased that du Sautoy was promoting mathematics, they were not always convinced by the way that he did it.

I myself expected to have a similar attitude to Finding Moonshine, but du Sautoy surprised me: he has pulled off that rare feat of writing in a way that can entertain and inform two different audiences – expert and non-expert – at the same time.

Okay, so maybe I should give ‘Finding Moonshine’ a further chance. After all, it is week-end and, I have nothing else to do than attending two family-parties… so I read the entire book in a couple of hours (not that difficult to do if you skip all paragraphs that have the look and feel of being copied from the books mentioned above) and, I admit, towards the end I mellowed a bit. Reading his diary notes I even felt empathy at times (if this is possible as du Sautoy makes a point of telling the world that most of us mathematicians are Aspergers). One example :

One of my graduate students has just left my office. He’s done some great work over the past three years and is starting to write up his doctorate, but he’s just confessed that he’s not sure that he wants to be a mathematician. I’m feeling quite sobered by this news. My graduate students are like my children. They are the future of the subject. Who’s going to read all the details of my papers if not my mathematical offspring? The subject feels so tribal that anyone who says they want out is almost a threat to everything the tribe stands for.
Anton has been working on a project very close to my current problem. There’s no denying that one can feel quite disillusioned by not finding a way into a problem. Last year one of my post-docs left for the City after attempting to scale this mountain with me. I’d already rescued him from being dragged off to the City once before. But after battling with our problem and seeing it become more and more complex, he felt that he wasn’t really cut out for it.

What is unsettling for me is that they both questioned the importance of what we are doing. They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.

Anton has questioned whether the problems we are working on are really important. I’ve explained why I think these are fundamental questions about basic objects in nature, but I can see that he isn’t convinced. I feel I am having to defend my whole existence. I’ve arranged for him to join me at a conference in Israel later this month, and I hope that seeing the rest of the tribe enthused and excited about these problems will re-inspire him. It will also show him that people are interested in what he is dedicating his time to.

Du Sautoy is a softy! I’d throw such students out of the window…

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Decryptable, only on fridays

The mini-post Bill Gates’ favourite prime number, encrypted below, can only be read on a friday. Here’s why.

[BEGIN URLCRYPT decode at lce.xamai.ca/urlcrypt.php]

cScSYXdhkQUWRwVOMHzMMFdHwVdCU3VU5LcSNgXc
1VB2plVn7jPqxmJD51UWFETGVWUTR2XNNmH89EBH
M3EYxUAKcxOPoEVwcjJgUCGX4Bdd0xTkAFW2YVGi
YoEqs1FulUWaZRQCalJRd0Ix0iNq4BnAdERckxfE
VEMYpRGwM1MUIlJSN0Sd2VVXETUdFTZwUDq6IEVe
Z1EZFDFVlUIDx0Y+ZRYXQyWCREGWs0RIVzPItGGh
MDLSZFEXZFWqAWXFpSVUN2VjU1R4FGVCdlFdlVLM
VUl8czJlZCWDBRXc9jFhoFRrYVGmAFZhplViMRXd
x0SOFTGVoVM10zNgAhUYl1RTglKUIEKZh1N8dFIU
sVLGZnM1ITz6UnXBcXc1FncGwQBJYgACQ3BJUHC1
tweDMnBAcnBEsQ3KkwBk0VUmcSJ8N9eQoAEDlzOP
ljQF3CELZCUpEUWhZlKdFlSIdRZLBVMkA4Al9CXe
J3EW5VMvRRHoEFVqEVIQZ115YFWYYVc98kM4c9Zr
12drRURBB0EJL1KVFBaZl0YeECjUQUNdFFTLZFWh
gNGZRSMqAiLRNlXrZUVWMyWRcDRQRC5bBCFSlyRQ
sUWYM/Wu0lQlZTJl9GXQbVXaB1UhUVShglWZYiWw
QRWoMxVWBZUMl1NXZVZ2ESIpbDXU5lWTIELUsgyo
EFVqEVIQZVOWJIWYYVUXATXXhCMgoSZwk1XE5FQe
9iF6lmYukXYklmcmxHdpNXauc3d39yLmrDc0RHaA
5FQTYV5TZDFEtSSZ0yVss0QXkCVZBFGL5lYMtEQp
ADIq4CESh1BHNBUqQxEoEFXzAvQlQxUzYlXagRX5
8lFa8DSPlEIuk1rFB0EUUEIAB1AQIbT20lJVdBOS
NEGTdFTXETUdFTZ3QSVrABWYR0EfgCQZNladt0Jf
cTWYYSQo+lFdVFW2gET3YiprpWeANERbtxaitkRU
lSUcFTUXQhU+3iUVpnGjdxOaVRXhAyLmICWXMkUE
dqFoEVR3kkSjdFLkD0FtJAAIoAGR1ymYkVKkISZm
g1QQkYXaZxKRlFMQ0kNjBnT50DTCUQAA4w5CoXAB
IXd2NneDAJAJsQBCEXDJMHBOAQdEM3BFM3CIoQyM
EAAxpQB0Znc8NrdIAAET4gF14DPF4UPRdzWzQha5
YlOEdxYIh2P+50JnEDLxISXr12SWJkHwQxQhYkVj
ZVIa5VmnYFVY0lTFdzWV0NJlNyKqMkQQw1RbZEPG
JFZV9UMHdyEUQFKHBUUURlUidlVaVWIhYjISdxQ+
IFRWASWUxyUqNW5GpSUaRCVRZVW1p1FxwUXiwiFl
9SUfNUWUplcldkFwYlqWBTX2clXMkTPy8aNFowfF
gAe4lHeH4XDK0gDOsAeJwQq51AB+9QeJoAfOkUDF
UQBKgUNHByJjmCMtAhUdpVQGVmGRVULCxELEViUX
EsZAVFTZ93FuQFXdcgC

[END URLCRYPT]

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the King’s problem on MUBs

MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk on MUBs (as well as an old talk by Wootters).

So what are MUBs to mathematicians? Recall that a d-state quantum system is just the vectorspace $\mathbb{C}^d $ equipped with the usual Hermitian inproduct $\vec{v}.\vec{w} = \sum \overline{v_i} w_i $. An observable $E $ is a choice of orthonormal basis ${ \vec{e_i} } $ consisting of eigenvectors of the self-adjoint matrix $E $. $E $ together with another observable $F $ (with orthonormal basis ${ \vec{f_j} } $) are said to be mutally unbiased if the norms of all inproducts $\vec{f_j}.\vec{e_i} $ are equal to $1/\sqrt{d} $. This definition extends to a collection of pairwise mutually unbiased observables. In a d-state quantum system there can be at most d+1 mutually unbiased bases and such a collection of observables is then called a MUB of the system. Using properties of finite fields one has shown that MUBs exists whenever d is a prime-power. On the other hand, existence of a MUB for d=6 still seems to be open…

The King’s Problem (( actually a misnomer, it’s more the poor physicists’ problem… )) is the following : A physicist is trapped on an island ruled by a mean
king who promises to set her free if she can give him the answer to the following puzzle. The
physicist is asked to prepare a d−state quantum system in any state of her choosing and give it
to the king, who measures one of several mutually unbiased observables on it. Following this, the physicist is allowed to make a control measurement
on the system, as well as any other systems it may have been coupled to in the preparation
phase. The king then reveals which observable he measured and the physicist is required
to predict correctly all the eigenvalues he found.

The Solution to the King’s problem in prime power dimension by P. K. Aravind, say for $d=p^k $, consists in taking a system of k object qupits (when $p=2l+1 $ one qupit is a spin l particle) which she will give to the King together with k ancilla qupits that she retains in her possession. These 2k qupits are diligently entangled and prepared is a well chosen state. The final step in finding a suitable state is the solution to a pure combinatorial problem :

She must use the numbers 1 to d to form $d^2 $ ordered sets of d+1 numbers each, with repetitions of numbers within a set allowed, such that any two sets have exactly one identical number in the same place in both. Here’s an example of 16 such strings for d=4 :

11432, 12341, 13214, 14123, 21324, 22413, 23142, 24231, 31243, 32134, 33421, 34312, 41111, 42222, 43333, 44444

Here again, finite fields are used in the solution. When $d=p^k $, identify the elements of $\mathbb{F}_{p^k} $ with the numbers from 1 to d in some fixed way. Then, the $d^2 $ of number-strings are found as follows : let $k_0,k_1 \in \mathbb{F}_{p^k} $ and take as the first 2 numbers the ones corresponding to these field-elements. The remaning d-2 numbers in the string are those corresponding to the field element $k_m $ (with $2 \leq m \leq d $) determined from $k_0,k_1 $ by the equation

$k_m = l_{m} * k_0+k_1 $

where $l_i $ is the field-element corresponding to the integer i ($l_1 $ corresponds to the zero element). It is easy to see that these $d^2 $ strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine $k_0,k_1 $ (and hence the whole string) as it follows from
$k_m = l_m k_0 + k_1 $ and $k_r = l_r k_0 + k_1 $ that $k_0 = \frac{k_m-k_r}{l_m-l_r} $.

In the special case when d=3 (that is, one spin 1 particle is given to the King), we recover the tetracode : the nine codewords

0000, 0+++, 0—, +0+-, ++-0, +-0+, -0-+, -+0-, –+0

encode the strings (with +=1,-=2,0=3)

3333, 3111, 3222, 1312, 1123, 1231, 2321, 2132, 2213

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