Terry Tao is reworking some of his better blogposts into a book, to be published by the AMS (here’s a preliminary version of the book “What’s New?”)

After some thought, I decided not to transcribe all of my posts from last year (there are 93 of them!), but instead to restrict attention to those articles which (a) have significant mathematical content, (b) are not announcements of material that will be published elsewhere, and (c) are not primarily based on a talk given by someone else. As it turns out, this still leaves about 33 articles from 2007, leading to a decent-sized book of a couple hundred pages in length.

If you have a blog and want to turn it into a LaTeX-book, there’s no need to transcribe or copy every single post, thanks to the WPTeX tool. Note that this is NOT a WP-plugin, but a (simple at that) php-program which turns all posts into a bookcontent.tex file. This file can then be edited further into a proper book.

Unfortunately, the present version chokes on LaTeXrender-code (which is easy enough to solve doing a global ‘find-and-replace’ of the tex-tags by dollar-signs) but worse, on Markdown-code… But then, someone fluent in php-regex will have no problems extending the libs/functions.php file (I hope…).

At the moment I’m considering turning the Mathieu-games-posts into a booklet. A possible title might be Mathieumatical Games. Rereading them (and other posts) I regret to be such an impatient blogger. Often I’m interested in something and start writing posts about it without knowing where or when I’ll land. This makes my posts a lot harder to get through than they might have been, if I would blog only after having digested the material myself… Typical recent examples are the tori-crypto-posts and the Bost-Connes algebra posts.

So, I still have a lot to learn from other bloggers I admire, such as Jennifer Ouellette who maintains the Coctail Party Physics blog. At the moment, Jennifer is resident blogger-journalist at the Kavli Institute where she is running a “Journal Club” workshop giving ideas on how to write better about science.

But the KITP is also committed to fostering scientific communication. That’s where I come in. Each Friday through April 26th, I’ll be presiding over a “Journal Club” meeting focusing on some aspect of communicating science.

Her most recent talk was entitled To Blog or Not to Blog? That is the Question and you can find the slides as well as a QuickTime movie of her talk. They even plan to set up a blog for the participants of the workshop. I will surely follow the rest of her course with keen interest!

We will see later that the cyclic subgroup is a 2-dimensional torus.

Take a finite set of polynomials and consider for every fieldextension the set of all k-tuples satisfying all these polynomials and call this set

Then, being a 2-dimensional torus roughly means that we can find a system of polynomials such that and over the algebraic closure we have and is a subgroup of this product group.

It is known that all 2-dimensional tori are rational. In particular, this means that we can write down maps defined by rational functions (fractions of polynomials) and which define a bijection between the points where f and j are defined (that is, possibly excluding zeroes of polynomials appearing in denumerators in the definition of the maps f or j). But then, we can use to map f to represent ‘most’ elements of by just 2 pits, exactly as in the XTR-system.

Making the rational maps f and j explicit and checking where they are ill-defined is precisely what Karl Rubin and Alice Silverberg did in their CEILIDH-system. The acronym CEILIDH (which they like us to pronounce as ‘cayley’) stands for Compact Efficient Improves on LUC, Improves on Diffie-Hellman…

A Cailidh is a Scots Gaelic word meaning ‘visit’ and stands for a ‘traditional Scottish gathering’.

Between 1997 and 2001 the Scottish ceilidh grew in popularity again amongst youths. Since then a subculture in some Scottish cities has evolved where some people attend ceilidhs on a regular basis and at the ceilidh they find out from the other dancers when and where the next ceilidh will be.
Privately organised ceilidhs are now extremely common, where bands are hired, usually for evening entertainment for a wedding, birthday party or other celebratory event. These bands vary in size, although are commonly made up of between 2 and 6 players. The appeal of the Scottish ceilidh is by no means limited to the younger generation, and dances vary in speed and complexity in order to accommodate most age groups and levels of ability.

Anyway, let us give the details of the Rubin-Silverberg approach. Take a large prime number p congruent to 2,6,7 or 11 modulo 13 and such that is again a prime number. Then, if is a 13-th root of unity we have that . Consider the elements

Then, for every define the map to by

and one can verify that this is indeed an element of provided we take

Conversely, for write using the basis , so and consequently write

with using the basis of . Okay, then the invers of iis the map given by

and it takes some effort to show that f and j are indeed each other inverses, that j is defined on all points of and that f is defined everywhere except at the two points . Therefore, as long as we avoid these two points in our Diffie-Hellman key exchange, we can perform it using just pits : I will send you allowing you to compute our shared key or from my data and your secret number b.

But, where’s the cat in all of this? Unfortunately, the cat is dead…

Yesterday, Jan Stienstra gave a talk at theARTS entitled “Quivers, superpotentials and Dimer Models”. He started off by telling that the talk was based on a paper he put on the arXiv Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d’Enfants but that he was not going to say a thing about dessins but would rather focuss on the connection with superpotentials instead…pleasing some members of the public, while driving others to utter despair.

Anyway, it gave me the opportunity to figure out for myself what dessins might have to do with dimers, whathever these beasts are. Soon enough he put on a slide containing the definition of a dimer and from that moment on I was lost in my own thoughts… realizing that a dessin d’enfant had to be a dimer for the Dedekind tessellation of its associated Riemann surface! and a few minutes later I could slap myself on the head for not having thought of this before :

There is a natural way to associate to a Farey symbol (aka a permutation representation of the modular group) a quiver and a superpotential (aka a necklace) defining (conjecturally) a Calabi-Yau algebra! Moreover, different embeddings of the cuboid tree diagrams in the hyperbolic plane may (again conjecturally) give rise to all sorts of arty-farty fanshi-wanshi dualities…

I’ll give here the details of the simplest example I worked out during the talk and will come back to general procedure later, when I’ve done a reference check. I don’t claim any originality here and probably all of this is contained in Stienstra’s paper or in some physics-paper, so if you know of a reference, please leave a comment. Okay, remember the Dedekind tessellation ?

So, all hyperbolic triangles we will encounter below are colored black or white. Now, take a Farey symbol and consider its associated special polygon in the hyperbolic plane. If we start with the Farey symbol

we get the special polygonal region bounded by the thick edges, the vertical edges are identified as are the two bottom edges. Hence, this fundamental domain has 6 vertices (the 5 blue dots and the point at ) and 8 hyperbolic triangles (4 colored black, indicated by a black dot, and 4 white ones).

Right, now let us associate a quiver to this triangulation (which embeds the quiver in the corresponding Riemann surface). The vertices of the triangulation are also the vertices of the quiver (so in our case we are going for a quiver with 6 vertices). Every hyperbolic edge in the triangulation gives one arrow in the quiver between the corresponding vertices. The orientation of the arrow is determined by the color of a triangle of which it is an edge : if the triangle is black, we run around its edges counter-clockwise and if the triangle is white we run over its edges clockwise (that is, the orientation of the arrow is independent of the choice of triangles to determine it). In our example, there is one arrows directed from the vertex at to the vertex at , whether you use the black triangle on the left to determine the orientation or the white triangle on the right. If we do this for all edges in the triangulation we arrive at the quiver below

where x,y and z are the three finite vertices on the -axis from bottom to top and where I’ve used the physics-convention for double arrows, that is there are two F-arrows, two G-arrows and two H-arrows. Observe that the quiver is of Calabi-Yau type meaning that there are as much arrows coming into a vertex as there are arrows leaving the vertex.

Now that we have our quiver we determine the superpotential as follows. Fix an orientation on the Riemann surface (for example counter-clockwise) and sum over all black triangles the product of the edge-arrows counterclockwise MINUS sum over all white triangles the product of the edge arrows counterclockwise. So, in our example we have the cubic superpotential

From this we get the associated noncommutative algebra, which is the quotient of the path algebra of the above quiver modulo the following ‘commutativity relations’

and morally this should be a Calabi-Yau algebra¹. This concludes the walk through of the procedure. Summarizing : to every Farey-symbol one associates a Calabi-Yau quiver and superpotential, possibly giving a Calabi-Yau algebra!

1. can someone who knows more about CYs verify this? [↩]

Just read/glanced through another math-for-the-masses book : The secret life of numbers by George G. Szpiro. The subtitle made me buy the book : 50 easy pieces on how mathematicians work and think Could be fun, I thought, certainly after reading the Amazon-blurb :

Most of us picture mathematicians laboring before a chalkboard, scribbling numbers and obscure symbols as they mutter unintelligibly. This lighthearted (but realistic) sneak-peak into the everyday world of mathematicians turns that stereotype on its head. Most people have little idea what mathematicians do or how they think. It’s often difficult to see how their seemingly arcane and esoteric work applies to our own everyday lives. But mathematics also holds a special allure for many people. We are drawn to its inherent beauty and fascinated by its complexity - but often intimidated by its presumed difficulty. \”The Secret Life of Numbers\” opens our eyes to the joys of mathematics, introducing us to the charming, often whimsical side, of the discipline.

Please correct me when I’m wrong, but I found just one out of 50 pieces which remotely fulfills this promise : ‘Cozy Zurich’¹. Still, there are some other pieces worth reading, 1. ‘A puzzle by any other name’² 2. ‘Twins, cousins and sexy primes’ ³ 3. ‘Proving the proof’⁴ 4. ‘Has Poincare’s conjecture finally been solved’⁵ 5. ‘Late tribute to a tragic hero’⁶ 6. ‘God’s gift to science?’⁷ to single out a few, embedded in a soup made out of the usual suspects (knots, chaos, RSA etc.). But, all in all, I fear the book doesn’t fulfill its promises and once again it demonstrates how little ‘math-substance’ one is able to put in a book for a general audience. But let us end with a quote from the preface that I really like :

Whenever a socialite shows off his flair at a coctail party by reciting a stanza from an obscure poem, he is considered well-read and full of wit. Not much ado can be made with the recitation of a mathematical formula, however. At most, one may expect a few pitying glances and the title ‘party’s most nerdy guest’. To the concurring nods of the cocktail crowd, most bystanders will admit that they are no good at math, never have been, and never will be.
Actually, this is quite astonishing. Imagine your lawyer telling you that he is no good at spelling, your dentist proudly proclaiming that she speaks n foreign language, and your financial advisor admitting with glee that he always mixes up Voltaire with Moliere. With ample reason one would consider such people as ignorant. Not so with mathematics. Shortcomings in this intellectual discipline are met with understanding by everyone.

1. on the awesome technical support a lecturer in Zurich is rumoured to receive [↩]
2. On the Collatz problem [↩]
3. How reasearch into the twin primes problem led to the discovery of a Pentium-bug [↩]
4. On Kepler’s problem [↩]
5. Of course it has been [↩]
6. On Abel’s life and prize [↩]
7. Stephen Wolfram bashing [↩]

The Category-Cafe ran an interesting post The history of n-categories claiming that “mathematicians’ histories are largely ‘Royal-road-to-me’ accounts”

To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently, while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.

Over the last weeks I’ve been writing up the notes of a course on ‘Elementary Algebraic Geometry’ that I’ll be teaching this year in Bach3. These notes are split into three historical periods more or less corresponding to major conceptual leaps in the subject : (1890-1920) ideals in polynomial rings (1920-1950) intrinsic definitions using the coordinate ring (1950-1970) scheme theory. Whereas it is clear to take Hilbert&Noether as the leading figures of the first period and Serre&Grothendieck as those of the last, the situation for the middle period is less clear to me. At first I went for the widely accepted story, as for example phrased by Miles Reid in the Final Comments to his Undergraduate Algebraic Geometry course.

… rigorous foundations for algebraic geometry were laid in the 1920s and 1930s by van der Waerden, Zariski and Weil (van der Waerden’s contribution is often suppressed, apparently because a number of mathematicians of the immediate post-war period, including some of the leading algebraic geometers, considered him a Nazi collaborator).

But then I read The Rising Sea: Grothendieck on simplicity and generality I by Colin McLarty and stumbled upon the following paragraph

From Emmy Noether’s viewpoint, then, it was natural to look at prime ideals instead of classical and generic points‚Äîor, as we would more likely say today, to identify points with prime ideals. Her associate Wolfgang Krull did this. He gave a lecture in Paris before the Second World War on algebraic geometry taking all prime ideals as points, and using a Zariski topology (for which see any current textbook on algebraic geometry). He did this over any ring, not only polynomial rings like C[x, y]. The generality was obvious from the Noether viewpoint, since all the properties needed for the definition are common to all rings. The expert audience laughed at him and he abandoned the idea.

The story seems to be due to Jurgen Neukirch’s ‘Erinnerungen an Wolfgang Krull’ published in ‘Wolfgang Krull : Gesammelte Abhandlungen’ (P. Ribenboim, editor) but as our library does not have this book I would welcome any additional information such as : when did Krull give this talk in Paris? what was its precise content? did he introduce the prime spectrum in it? and related to this : when and where did Zariski introduce ‘his’ topology? Answers anyone?