Category: stories

un-doing the Grothendieck?

Published January 23, 2008 by lievenlb

(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!

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now what?

Published January 14, 2008 by lievenlb

You may not have noticed, but the really hard work was done behind the scenes, resurrecting about 300 old posts (some of them hidden by giving them ‘private’-status). Ive only deleted about 10 posts with little or no content and am sorry I’ve self-destructed about 20-30 hectic posts over the years by pressing the ‘delete post’ button. I would have liked to reread them after all the angry mails Ive received. But, as Ive defended myself at the time, and as I continue to do today, a blog only records feelings at a specific moment. Often, the issue is closed for me once Ive put my frustrations in a post, and then Ill forget all about it. Sadly, the gossip-circuit in noncommutative circles is a lot, a lot, slower than my mood swings, so by the time people complain it’s no longer an issue for me and I tend to delete the post altogether. A blog really is a sort of diary. For example, it only struck me now, rereading the posts of the end of 2006, beginning of 2007, how depressed I must have been at the time. Fortunately, life has improved, somewhat… Still, after all these reminiscences, the real issue is : what comes next?

Some of you may have noticed that I’ve closed the open series on tori-cryptography and on superpotentials in a rather abrupt manner. It took me that long to realize that none of you is waiting for this kind of posts. You’re thinking : if he really wants to show off, let him do his damned thing on the arXiv, a couple of days a year, at worst, and then we can then safely ignore it, like we do with most papers. Isnt’t that true? Of course it is…

So, what are you waiting for? Here’s what I believe to be a sensible thing to try out. Over the last 4 years I must have posted well over 50 times what I believe noncommutative geometry is all about, so if you still don’t know, please consult the archive, I fear I can only repeat myself. Probably, it is more worthwhile to reach out to other approaches to noncommutative geometry, trying to figure out what, if anything, they are after, without becoming a new-age convert (‘connes-vert’, I’d say). The top-left picture may give you an inkling of what I’m after… Besides, Im supposed to run a ‘capita selecta’ course for third year Bachelors and Ive chosen to read with them the book The music of the primes and to expand on the mathematics hinted only at in the book. So, I’ll totally immerse myself in Connes’ project to solve the Riemann-hypothesis in the upcoming months.

Again, rereading old posts, it strikes me how much effort I’ve put into trying to check whether technology can genuinely help mathematicians to do what they want to do more efficiently (all post categorized as iMath). I plan some series of posts re-exploring these ideas. The first series will be about the overhyped Web-2 thing of social-bookmarking. So, in the next weeks I’ll go undercover and check out which socialsites are best for mathematicians (in particular, noncommutative geometers) to embrace…

Apart from these, admittedly vague, plans I am as always open for suggestions you might have. So, please drop a comment..

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microtrends & mathematics

Published January 13, 2008 by lievenlb

Mark J. Penn wrote Microtrends: The Small Forces Changing the World. He argues that the most important trends in the world today are the smallest ones. Such as… declining standards in math education!

What should you do on the educational front if you have a child with an aptitude for numbers, as mine does? Both of you had better get cracking, because American college students are studying less math. As an example, “Microtrends” says Harvard has only 77 math majors out of 6,700 undergraduate students.

The math story is different in China and India, which are graduating as many as 950,000 engineers a year. Granted, both nations are far more populous than the United States, but that is a lot of engineers.

Mr. Penn notes that a 2001 bipartisan commission “said that the greatest threat to American national security – behind only terrorist attacks – was the threat of failing to provide sufficient math and science education in America.”

I haven’t read the book yet but it’s high on my wish-list after reading the NYT-article Why There’s Strength in Small Numbers and the Introduction of the book.

Doodles worth millions (or not)

Published January 10, 2008 by lievenlb

Via PD1, who told me the story on her 23rd birthday, yesterday.

The story starts with Alex Matter, whose father, Herbert, and mother, Mercedes, were artists and friends of Jackson Pollock, famous for his drip-paintings. He discovered a group of small drip paintings in a storage locker in Wainscott, N.Y. which he believed to be authentic Pollocks, and if he is proved right, they would be worth millions of dollars.

Usually such discoveries lead to heated debates among art-critics and Pollock-experts whether one finds proof to authenticate the paintings. Not this time. In steps a mathematician who claims that he can authenticate a Pollock drip-painting by calculating its fractal dimension (??!!)… and claims that these drippings cannot be Pollocks because their dimension is too small… LOL!

This madmatician is Richard Taylor from the University of Oregon in Corvallis.

Taylor took a digital image of a Pollock painting into his lab, broke the image into its separate colors, and computed the fractal dimension of the lines in each color. Each time, he got a number between 1 and 2, confirming his notion that Pollock’s paintings are fractal. “Rather than mimicking nature,” Taylor says, Pollock “adopted its language of fractals to build his own patterns.”

In 1999, Taylor reported that the fractal dimension of Pollock’s paintings increased during his life. His early drip paintings have a loose web of lines, mostly at the same scale. Because these paintings show no fractal qualities, their dimension is near 1. But Pollock’s later paintings have a dense network of overlapping lines, ranging from large, bold strokes to delicate threads, Taylor calculated a fractal dimension of 1.72 for these works.

His paper on this “Authenticating Pollock Paintings Using Fractal Geometry” can be found here. Luckily, the story doesn’t end here. In steps a graduate student in astrophysics at Case Western, Katherine Jones-Smith who had to give a seminar talk to her fellow students.

“I was sort of bored with particle astrophysics,” Jones-Smith says, so she looked around for something different. She came across an account of Taylor’s work, and “it sounded really cool,” she recalls.

“The obvious check to me was to make sure that not any old scribble would appear to be fractal,” she says. “So, I made some scribbles.” Much to her surprise, when she computed the fractal dimension of her scribbles, they turned out to be greater than 1.

Recently, she arXived her findings in the paper Drip Paintings and Fractal Analysis from which the above doodles are taken, called appropriately “Gross pebbles” and “Mixed stars”.

When Katherine Jones-Smith made some doodles on a page “”pretty ugly” ones, she says”she found that they shared the qualities of a Pollock, according to an analysis that follows Taylor’s approach. “Either Taylor is wrong, or Kate’s drawings are worth $40 million,” says Jones-Smith’s collaborator Harsh Mathur. “We’d be happy either way.“

More on this hilarious story can be found in this science news article, this New-York times story or the Pollocks-bollocks blog entry over at biophemera.

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sobering-up

Published January 5, 2008 by lievenlb

Kea’s post reminded me to have a look at my search terms (the things people type into search engines to get redirected here). Quite a sobering experience…

Via Google Analytics I learn that 49,51% of traffic comes from Search Engines (compared to 26,17% from Referring Sites and 24,32% from direct hits) so I should take Search Terms more seriously! Above you can find the top-25.

On 1. there is neverendingbooks. Well, some people seem to remember the blog-name, but require google to remember the URL (neverendingbooks.org)…, okay, fair enough. But from then on… all search terms are iTouch related! The first ‘other’ term is puzzle m at 24. and believe me things do not improve afterwards. Here the only non-Touch related search terms in the top 100 :

neverendingbooks.org (40)
“puzzle m” (42)
moonshine mathematics (79)
necklace algebra (80)
“calabi-yau algebra (90)
“dessin d enfant” (91)
“lieven le bruyn” (95)
Mathieu group + M(13) (97)
13 points 5 lines puzzle (98)
15 itouch sliding puzzle (99)

the last one is really touching (sic). Is there anybody out there still interested in the mathematics, or should I turn this blog into a yaib (yet another iTouch blog) ???

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mathematics for 2008 (and beyond)

Published December 30, 2007 by lievenlb

Via the n-category cafe (and just now also the Arcadian functor ) I learned that Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century.

DARPA is the “Defense Advanced Research Projects Agency” and is an agency of the United States Department of Defense ‘responsible for the development of new technology for use by the military’.

Bejamin Mann is someone in their subdivision DSO, that is, the “Defense Sciences Office” that ‘vigorously pursues the most promising technologies within a broad spectrum of the science and engineering research communities and develops those technologies into important, radically new military capabilities’.

I’m not the greatest fan of the US military, but the proposed list of 23 mathematical challenges is actually quite original and interesting.

What follows is my personal selection of what I consider the top 5 challenges from the list (please disagree) :

1. The Mathematics of Quantum Computing, Algorithms, and Entanglement (DARPA 15) : “In the last century we learned how quantum phenomena shape
our world. In the coming century we need to develop the
mathematics required to control the quantum world.”

2. Settle the Riemann Hypothesis (DARPA 19) : “The Holy Grail of number theory.”

3. Geometric Langlands and Quantum Physics (DARPA 17) : “How does the Langlands program, which originated in number
theory and representation theory, explain the fundamental
symmetries of physics? And vice versa?”

4. The Geometry of Genome Space (DARPA 15) : “What notion of distance is needed to incorporate biological utility?”

5. Algorithmic Origami and Biology (DARPA 10) : “Build a stronger mathematical theory for isometric and rigid
embedding that can give insight into protein folding.”

All of this will have to wait a bit, for now

HAPPY & HEALTHY 2008

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SMS-Math Meme (SMM)

Published December 21, 2007 by lievenlb

Hey, here’s an idea : The Text-Math Book! Trying to promote mathematics while at the same time acknowledging the fairly limited attention-span of the intended generation, let’s try to write a book on serious maths following just one rule

EVERY DEFINITION, THEOREM AND PROOF IN THE BOOK SHOULD NOT BE LONGER THAN A TEXT-MESSAGE (ie. 160 chars)

I don’t even own a cell phone (( waiting for the iPhone to arrive in Belgium )), so PLEASE educate me youngsters! SMS your contribution, either as a comment left here or hosted at your own blog (please link, so that I can learn…, a full text explanation of abbreviations used will be applauded.)

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daddy wasn’t impressed

Published December 19, 2007 by lievenlb

A first year-first semester course on group theory has its hilarious moments. Whereas they can relate the two other pure math courses (linear algebra and analysis) _somewhat_ to what they’ve learned before, with group theory they appear to enter an entirely new and strange world. So, it is best to give them concrete examples : symmetry groups of regular polygons and Platonic solids, the symmetric group etc. One of the lesser traditional examples I like to give is Nim addition and its relation to combinatorial games.

For their first test they had (among other things) to find a winning move for the position below in the Lenstra’s turtle turning game. At each move a player must put one turtle on its back and may also turn over any single turtle to the left of it. This second turtle, unlike the first, may be turned either onto its feet or onto its back. The player wins who turns the last turtle upside-down.

So, all they needed to see was that one turtle on its feet at place n is equivalent to a Nim-heap of height n and use the fact that all elements have order two to show that any zero-move in the sum game can indeed be played by using the second-turtle alternative. (( for the curious : the answer is turning both 9 and 4 on their back ))

A week later, one of the girls asked at the start of the lecture :

Are there real-life applications of group-theory? I mean, my father asked me what I was learning at school and I told him we were playing games turning turtles. I have to say that he was not impressed at all!.

She may have had an hidden agenda to slow me down because I spend an hour talking about a lot of things ranging from codes to cryptography and from representations to elementary particles…

For test three (on group-actions) I asked them to prove (among other things) Wilson’s theorem that is

$~(p-1)! \equiv -1~\text{mod}~p $

for any prime number $p $. The hint being : take the trivial action of $S_p $ on a one-element set and use the orbit theorem. (they know the number of elements in an $S_n $-conjugacy class)

Fingers crossed, hopefully daddy approved…

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problema bovinum

Published December 3, 2007 by lievenlb

Suppose for a moment that some librarian at the Bodleian Library announces that (s)he discovered an old encrypted book attributed to Isaac Newton. After a few months of failed attempts, the code is finally cracked and turns out to use a Public Key system based on the product of two gigantic prime numbers, $2^{32582657}-1 $ and $2^{30402457}-1 $, which were only discovered to be prime recently. Would one deduce from this that Newton invented public key cryptography and that he used alchemy to factor integers? (( Come to think of it, some probably would ))

The cynic in me would argue that it is a hell of a coincidence for this text to surface exactly at the moment in history when we are able to show these numbers to be prime and understand their cryptographic use, and conclude that the book is likely to be a fabrication. Still, stranger things have happened in the history of mathematics…

In 1773, Gotthold Ephraim Lessing at that time librarian at the Herzog-August-Bibliothek discovered and published a Greek epigram in 22 elegiac couplets. The manuscript describes a problem sent by Archimedes to the mathematicians in Alexandria.

In his beautiful book “Number Theory, an approach through history. From Hammurapi to Legendre” Andre Weil asserts (( Chapter I,IX )):

Many mathematical epigrams are known. Most of them state problems of little depth; not so Lessing’s find; there is indeed every reason to accept the attribution to Archimedes, and none for putting it into doubt.

This Problema Bovidum (the cattle problem) is a surprisingly difficult diophantine problem and the simplest complete solution consists of eigth numbers, each having about 206545 digits. As we will see later the final ingredient in the solution is the solution of Pell’s equation using continued fractions discovered by Lagrange in 1768 and published in 1769 in a long memoir. Lagrange’s solution to the Pell equation was inserted in Euler’s “Algebra” which was composed in 1771 but published only in 1773… the very same year as Lessing’s discovery! (( all dates learned from Weil’s book Chp. III,XII ))

Weil’s book doesn’t include the details of the original epigram. The (lost) archeologist in me wanted to see the original Greek 22 couplets as well as a translation. So here they are : (( thanks to the Cattle problem site ))

A PROBLEM

which Archimedes solved in epigrams, and which he communicated to students of such matters at Alexandria in a letter to Eratosthenes of Cyrene.

If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.

The Lessing epigram may very well be an extremely laborious hoax but it is still worth spending a couple of posts on it. It gives us the opportunity to retell the amazing history of Pell’s problem rangingfrom the ancient Greeks and Indians, over Fermat and his correspondents, to Euler and Lagrange (with a couple of recent heroes entering the story). And, on top of this, the modular group is all the time just around the corner…

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Vacation reading

Published July 26, 2007 by lievenlb

Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal.

These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic Geometry and partly on David Mumford’s Red Book, but this turned out to be too heavy going. Next year I’ll be happy if they know enough on algebraic curves. The backbone of these two courses will be Fulton’s old but excellent Algebraic curves. It’s self contained (unlike Hartshorne’s book that assumes a prior course on commutative algebra), contains a lot of exercises and goes on to the Brill-Noether proof of Riemann-Roch. Still, Id like to extend it with the introductory chapter and the chapters on Riemann surfaces from Complex Algebraic Curves by Frances Kirwan, a bit on elliptic and modular functions from Elliptic curves by Henry McKean and Victor Moll and the adelic proof of Riemann-Roch and applications of it to the construction of algebraic codes from Algebraic curves over finite fields by Carlos Moreno. If time allows Id love to include also the chapter on zeta functions but I fear this will be difficult.

These are to spice up a 2nd year Bachelor course on Representations of Finite Groups with a tiny bit of Galois representations, both as motivation and to wet their appetite for elliptic curves and algebraic geometry. Ive received Fearless Symmetry by Avner Ash and Robert Gross only yesterday and find it hard to stop reading. It attempts to explain Galois representations and generalized reciprocity laws to a general audience and from what I read so far, they really do a terrific job. Another excellent elementary introduction to elliptic curves and Galois representations is in Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch. On a gossipy note, the appendix “The origin of the elliptic approach to Fermat’s last theorem” is fun reading. Finally, Ill also take Introduction to Fermat’s Last Theorem by Alf van der Poorten along simply because I love his writing style.

These are included just for fun. The Poincare Conjecture by Donal O’Shea because I know far too little about it, Letters to a Young Mathematician by Ian Stewart because I like the concept of the book and finally The sensual (quadratic) form by John Conway because I need to have at all times at least one Conway-book nearby.

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