Tag: geometry

The F_un folklore

Published June 7, 2008 by lievenlb

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries of the Priory of Sion, a secret society founded in the middle ages and still active today…

The followers of F-un, for $\mathbb{F}_1 $ the field of one element, have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.

The first are the lecture notes “Lectures on zeta functions and motives” by Yuri I. Manin of a course given in 1991.

One can download a scanned version of the paper from the homepage of Katia Consani as a huge 23.1 Mb file. Of F-un relevance is the first section “Absolute Motives?” in which

“…we describe a highly speculative picture of analogies between arithmetics over $\mathbb{F}_q $ and over $\mathbb{Z} $, cast in the language reminiscent of Grothendieck’s motives. We postulate the existence of a category with tensor product $\times $ whose objects correspond not only to the divisors of the Hasse-Weil zeta functions of schemes over $\mathbb{Z} $, but also to Kurokawa’s tensor divisors. This neatly leads to teh introduction of an “absolute Tate motive” $\mathbb{T} $, whose zeta function is $\frac{s-1}{2\pi} $, and whose zeroth power is “the absolute point” which is teh base for Kurokawa’s direct products. We add some speculations about the role of $\mathbb{T} $ in the “algebraic geometry over a one-element field”, and in clarifying the structure of the gamma factors at infinity.” (loc.cit. p 1-2)

I’d welcome links to material explaining this section to people knowing no motives.

The second one is the unpublished paper “Cohomology determinants and reciprocity laws : number field case” by Mikhail Kapranov and A. Smirnov.

This paper features in blog-posts at the Arcadian Functor, in John Baez’ Weekly Finds and in yesterday’s post at Noncommutative Geometry.

You can download every single page (of 15) as a separate file from here. But, in order to help spreading the Fun-gospel, I’ve made these scans into a single PDF-file which you can download as a 2.6 Mb PDF. In the introduction they say :

“First of all, it is an old idea to interpret combinatorics of finite sets as the $q \rightarrow 1 $ limit of linear algebra over the finite field $\mathbb{F}_q $. This had lead to frequent consideration of the folklore object $\mathbb{F}_1 $, the “field with one element”, whose vector spaces are just sets. One can postulate, of course, that $\mathbf{spec}(\mathbb{F}_1) $ is the absolute point, but the real problem is to develop non-trivial consequences of this point of view.”

They manage to deduce higher reciprocity laws in class field theory within the theory of $\mathbb{F}_1 $ and its field extensions $\mathbb{F}_{1^n} $. But first, let us explain how they define linear algebra over these absolute fields.

Here is a first principle : in doing linear algebra over these fields, there is no additive structure but only scalar multiplication by field elements. So, what are vector spaces over the field with one element? Well, as scalar multiplication with 1 is just the identity map, we have that a vector space is just a set. Linear maps are just set-maps and in particular, a linear isomorphism of a vector space onto itself is a permutation of the set. That is, linear algebra over $\mathbb{F}_1 $ is the same as combinatorics of (finite) sets.

A vector space over $\mathbb{F}_1 $ is just a set; the dimension of such a vector space is the cardinality of the set. The general linear group $GL_n(\mathbb{F}_1) $ is the symmetric group $S_n $, the identification via permutation matrices (having exactly one 1 in every row and column)

Some people prefer to view an $\mathbb{F}_1 $ vector space as a pointed set, the special element being the ‘origin’ $0 $ but as $\mathbb{F}_1 $ doesnt have a zero, there is also no zero-vector. Still, in later applications (such as defining exact sequences and quotient spaces) it is helpful to have an origin. So, let us denote for any set $S $ by $S^{\bullet} = S \cup { 0 } $. Clearly, linear maps between such ‘extended’ spaces must be maps of pointed sets, that is, sending $0 \rightarrow 0 $.

The field with one element $\mathbb{F}_1 $ has a field extension of degree n for any natural number n which we denote by $\mathbb{F}_{1^n} $ and using the above notation we will define this field as :

$\mathbb{F}_{1^n} = \mu_n^{\bullet} $ with $\mu_n $ the group of all n-th roots of unity. Note that if we choose a primitive n-th root $\epsilon_n $, then $\mu_n \simeq C_n $ is the cyclic group of order n.

Now what is a vector space over $\mathbb{F}_{1^n} $? Recall that we only demand units of the field to act by scalar multiplication, so each ‘vector’ $\vec{v} $ determines an n-set of linear dependent vectors $\epsilon_n^i \vec{v} $. In other words, any $\mathbb{F}_{1^n} $-vector space is of the form $V^{\bullet} $ with $V $ a set of which the group $\mu_n $ acts freely. Hence, $V $ has $N=d.n $ elements and there are exactly $d $ orbits for the action of $\mu_n $ by scalar multiplication. We call $d $ the dimension of the vectorspace and a basis consists in choosing one representant for every orbits. That is, $~B = { b_1,\ldots,b_d } $ is a basis if (and only if) $V = { \epsilon_n^j b_i~:~1 \leq i \leq d, 1 \leq j \leq n } $.

So, vectorspaces are free $\mu_n $-sets and hence linear maps $V^{\bullet} \rightarrow W^{\bullet} $ is a $\mu_n $-map $V \rightarrow W $. In particular, a linear isomorphism of $V $, that is an element of $GL_d(\mathbb{F}_{1^n}) $ is a $\mu_n $ bijection sending any basis element $b_i \rightarrow \epsilon_n^{j(i)} b_{\sigma(i)} $ for a permutation $\sigma \in S_d $.

An $\mathbb{F}_{1^n} $-vectorspace $V^{\bullet} $ is a free $\mu_n $-set $V $ of $N=n.d $ elements. The dimension $dim_{\mathbb{F}_{1^n}}(V^{\bullet}) = d $ and the general linear group $GL_d(\mathbb{F}_{1^n}) $ is the wreath product of $S_d $ with $\mu_n^{\times d} $, the identification as matrices with exactly one non-zero entry (being an n-th root of unity) in every row and every column.

This may appear as a rather sterile theory, so let us give an extremely important example, which will lead us to our second principle for developing absolute linear algebra.

Let $q=p^k $ be a prime power and let $\mathbb{F}_q $ be the finite field with $q $ elements. Assume that $q \cong 1~mod(n) $. It is well known that the group of units $\mathbb{F}_q^{\ast} $ is cyclic of order $q-1 $ so by the assumption we can identify $\mu_n $ with a subgroup of $\mathbb{F}_q^{\ast} $.

Then, $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} $ is an $\mathbb{F}_{1^n} $-vectorspace of dimension $d=\frac{q-1}{n} $. In other words, $\mathbb{F}_q $ is an $\mathbb{F}_{1^n} $-algebra. But then, any ordinary $\mathbb{F}_q $-vectorspace of dimension $e $ becomes (via restriction of scalars) an $\mathbb{F}_{1^n} $-vector space of dimension $\frac{e(q-1)}{n} $.

Next time we will introduce more linear algebra definitions (including determinants, exact sequences, direct sums and tensor products) in the realm the absolute fields $\mathbb{F}_{1^n} $ and remarkt that we have to alter the known definitions as we can only use the scalar-multiplication. To guide us, we have the second principle : all traditional results of linear algebra over $\mathbb{F}_q $ must be recovered from the new definitions under the vector-space identification $\mathbb{F}_q = (\mathbb{F}_q^{\ast})^{\bullet} = \mathbb{F}_{1^n} $ when $n=q-1 $. (to be continued)

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Looking for F_un

Published June 3, 2008 by lievenlb

There are only a handful of human activities where one goes to extraordinary lengths to keep a dream alive, in spite of overwhelming evidence : religion, theoretical physics, supporting the Belgian football team and … mathematics.

In recent years several people spend a lot of energy looking for properties of an elusive object : the field with one element $\mathbb{F}_1 $, or in French : “F-un”. The topic must have reached a level of maturity as there was a conference dedicated entirely to it : NONCOMMUTATIVE GEOMETRY AND GEOMETRY OVER THE FIELD WITH ONE ELEMENT.

In this series I’d like to find out what the fuss is all about, why people would like it to exist and what it has to do with noncommutative geometry. However, before we start two remarks :

The field $\mathbb{F}_1 $ does not exist, so don’t try to make sense of sentences such as “The ‘field with one element’ is the free algebraic monad generated by one constant (p.26), or the universal generalized ring with zero (p.33)” in the wikipedia-entry. The simplest proof is that in any (unitary) ring we have $0 \not= 1 $ so any ring must contain at least two elements. A more highbrow version : the ring of integers $\mathbb{Z} $ is the initial object in the category of unitary rings, so it cannot be an algebra over anything else.

The second remark is that several people have already written blog-posts about $\mathbb{F}_1 $. Here are a few I know of : David Corfield at the n-category cafe and at his old blog, Noah Snyder at the secret blogging seminar, Kea at the Arcadian functor, AC and K. Consani at Noncommutative geometry and John Baez wrote about it in his weekly finds.

The dream we like to keep alive is that we will prove the Riemann hypothesis one fine day by lifting Weil’s proof of it in the case of curves over finite fields to rings of integers.

Even if you don’t know a word about Weil’s method, if you think about it for a couple of minutes, there are two immediate formidable problems with this strategy.

For most people this would be evidence enough to discard the approach, but, we mathematicians have found extremely clever ways for going into denial.

The first problem is that if we want to think of $\mathbf{spec}(\mathbb{Z}) $ (or rather its completion adding the infinite place) as a curve over some field, then $\mathbb{Z} $ must be an algebra over this field. However, no such field can exist…

No problem! If there is no such field, let us invent one, and call it $\mathbb{F}_1 $. But, it is a bit hard to do geometry over an illusory field. Christophe Soule succeeded in defining varieties over $\mathbb{F}_1 $ in a talk at the 1999 Arbeitstagung and in a more recent write-up of it : Les varietes sur le corps a un element.

We will come back to this in more detail later, but for now, here’s the main idea. Consider an existent field $k $ and an algebra $k \rightarrow R $ over it. Now study the properties of the functor (extension of scalars) from $k $-schemes to $R $-schemes. Even if there is no morphism $\mathbb{F}_1 \rightarrow \mathbb{Z} $, let us assume it exists and define $\mathbb{F}_1 $-varieties by requiring that these guys should satisfy the properties found before for extension of scalars on schemes defined over a field by going to schemes over an algebra (in this case, $\mathbb{Z} $-schemes). Roughly speaking this defines $\mathbb{F}_1 $-schemes as subsets of points of suitable $\mathbb{Z} $-schemes.

But, this is just one half of the story. He adds to such an $\mathbb{F}_1 $-variety extra topological data ‘at infinity’, an idea he attributes to J.-B. Bost. This added feature is a $\mathbb{C} $-algebra $\mathcal{A}_X $, which does not necessarily have to be commutative. He only writes : “Par ignorance, nous resterons tres evasifs sur les proprietes requises sur cette $\mathbb{C} $-algebre.”

The algebra $\mathcal{A}_X $ originates from trying to bypass the second major obstacle with the Weil-Riemann-strategy. On a smooth projective curve all points look similar as is clear for example by noting that the completions of all local rings are isomorphic to the formal power series $k[[x]] $ over the basefield, in particular there is no distinction between ‘finite’ points and those lying at ‘infinity’.

The completions of the local rings of points in $\mathbf{spec}(\mathbb{Z}) $ on the other hand are completely different, for example, they have residue fields of different characteristics… Still, local class field theory asserts that their quotient fields have several common features. For example, their Brauer groups are all isomorphic to $\mathbb{Q}/\mathbb{Z} $. However, as $Br(\mathbb{R}) = \mathbb{Z}/2\mathbb{Z} $ and $Br(\mathbb{C}) = 0 $, even then there would be a clear distinction between the finite primes and the place at infinity…

Alain Connes came up with an extremely elegant solution to bypass this problem in Noncommutative geometry and the Riemann zeta function. He proposes to replace finite dimensional central simple algebras in the definition of the Brauer group by AF (for Approximately Finite dimensional)-central simple algebras over $\mathbb{C} $. This is the origin and the importance of the Bost-Connes algebra.

We will come back to most of this in more detail later, but for the impatient, Connes has written a paper together with Caterina Consani and Matilde Marcolli Fun with $\mathbb{F}_1 $ relating the Bost-Connes algebra to the field with one element.

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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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“God given time”

Published February 20, 2008 by lievenlb

If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it.

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure
theory, which in essence only cares about the “quantity of points” in a space, one
ﬁnds unexpected completely new features in the noncommutative case. While it
had been long known by operator algebraists that the theory of von-Neumann
algebras represents a far reaching extension of measure theory, the main surprise
which occurred at the beginning of the seventies is that such an algebra M
inherits from its noncommutativity a god-given time evolution:

$\delta~:~\mathbb{R} \rightarrow Out(M) $

where $Out M = Aut M/Int M $ is the quotient of the group of automorphisms of M
by the normal subgroup of inner automorphisms. This led in my thesis to the
reduction from type III to type II and their automorphisms and eventually to the
classiﬁcation of injective factors.

Even a commutative manifold has a kind of dynamics associated to it. Take a suitable vectorfield, consider the flow determined by it and there’s your ‘dynamics’, or a one-parameter group of automorphisms on the functions. Further, other classes of noncommutative algebras have similar features. For example, Cuntz and Quillen showed that also formally smooth algebras (the noncommutative manifolds in the algebraic world) have natural Yang-Mills flows associated to them, giving a one-parameter subgroup of automorphisms.

Let us try to keep far from mysticism and let us agree that by ‘time’ (let alone ‘god given time’) we mean a one-parameter subgroup of algebra automorphisms of the noncommutative algebra. In nice cases, such as some von-Neumann algebras this canonical subgroup is canonical in the sense that it is unique upto inner automorphisms.

In the special case of the Bost-Connes algebra these automorphisms $\sigma_t $ are given by $\sigma_t(X_n) = n^{it} X_n $ and $\sigma_t(Y_{\lambda}) = Y_{\lambda} $.

This one-parameter subgroup is crucial in the definition of the so called KMS-states (for Kubo-Martin and Schwinger) which is our next goal.

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censured post : bloggers’ block

Published February 6, 2008 by lievenlb

Below an up-till-now hidden post, written november last year, trying to explain the long blog-silence at neverendingbooks during october-november 2007…

A couple of months ago a publisher approached me, out of the blue, to consider writing a book about mathematics for the general audience (in Dutch (?!)). Okay, I brought this on myself hinting at the possibility in this post

Recently, I’ve been playing with the idea of writing a book for the general public. Its title is still unclear to me (though an idea might be “The disposable science”, better suggestions are of course wellcome) but I’ve fixed the subtitle as “Mathematics’ puzzling fall from grace”. The book’s concept is simple : I would consider the mathematical puzzles creating an hype over the last three centuries : the 14-15 puzzle for the 19th century, Rubik’s cube for the 20th century and, of course, Sudoku for the present century.

For each puzzle, I would describe its origin, the mathematics involved and how it can be used to solve the puzzle and, finally, what the differing quality of these puzzles tells us about mathematics’ changing standing in society over the period. Needless to say, the subtitle already gives away my point of view. The final part of the book would then be more optimistic. What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?

While I still like the idea and am considering the proposal, chances are low this book ever materializes : the blog-title says it all…

Then, about a month ago I got some incoming links from a variety of Flemish blogs. From their posts I learned that the leading Science-magazine for the low countries, Natuur, Wetenschap & Techniek (Nature, Science & Technology), featured an article on Flemish science-blogs and that this blog might be among the ones covered. It sure would explain the publisher’s sudden interest. Of course, by that time the relevant volume of NW&T was out of circulation so I had to order a backcopy to find out what was going on. Here’s the relevant section, written by their editor Erick Vermeulen (as well as an attempt to translate it)

Sliding puzzle For those who want more scientific depth (( their interpretation, not mine )), there is the English blog by Antwerp professor algebra & geometry Lieven Le Bruyn, MoonshineMath (( indicates when the article was written… )). Le Bruyn offers a number of mathematical descriptions, most of them relating to group theory and in particular the so called monster-group and monstrous moonshine. He mentions some puzzles in passing such as the well known sliding puzzle with 15 pieces sliding horizontally and vertically in a 4 by 4 matrix. Le Bruyn argues that this ’15-puzzle (( The 15-puzzle groupoid ))’ was the hype of the 19th century as was the Rubik cube for the 20th and is Sudoku for the 21st century.
Interesting is Le Bruyn’s mathematical description of the M(13)-puzzle (( Conway’s M(13)-puzzle )) developed by John Conway. It has 13 points on a circle, twelve of them carrying a numbered counter. Every point is connected via lines to all others (( a slight simplification )). Whenever a counter jumps to the empty spot, two others exchange places. Le Bruyn promises the blog-visitor new variants to come (( did I? )). We are curious.
Of course, the genuine puzzler can leave all this theory for what it is, use the Java-applet (( Egner’s M(13)-applet )) and painfully try to move the counters around the circle according to the rules of the game.

Some people crave for this kind of media-attention. On me it merely has a blocking-effect. Still, as the end of my first-semester courses comes within sight, I might try to shake it off…

vacation reading (2)

Published February 5, 2008 by lievenlb

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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the Bost-Connes coset space

Published January 17, 2008 by lievenlb

By now, everyone remotely interested in Connes’ approach to the Riemann hypothesis, knows the _one line mantra_

one can use noncommutative geometry to extend Weil’s proof of the Riemann-hypothesis in the function field case to that of number fields

But, can one go beyond this sound-bite in a series of blog posts? A few days ago, I was rather optimistic, but now, after reading-up on the Connes-Consani-Marcolli project, I feel overwhelmed by the sheer volume of their work (and by my own ignorance of key tools in the approach). The most recent account takes up half of the 700+ pages of the book Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli…

So let us set a more modest goal and try to understand one of the first papers Alain Connes wrote about the RH : Noncommutative geometry and the Riemann zeta function. It is only 24 pages long and relatively readable. But even then, the reader needs to know about class field theory, the classification of AF-algebras, Hecke algebras, etc. etc. Most of these theories take a book to explain. For example, the first result he mentions is the main result of local class field theory which appears only towards the end of the 200+ pages of Jean-Pierre Serre’s Local Fields, itself a somewhat harder read than the average blogpost…

Anyway, we will see how far we can get. Here’s the plan : I’ll take the heart-bit of their approach : the Bost-Connes system, and will try to understand it from an algebraist’s viewpoint. Today we will introduce the groups involved and describe their cosets.

For any commutative ring $R $ let us consider the group of triangular $2 \times 2 $ matrices of the form

$P_R = { \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix}~|~b \in R, a \in R^* } $

(that is, $a $ in an invertible element in the ring $R $). This is really an affine group scheme defined over the integers, that is, the coordinate ring

$\mathbb{Z}[P] = \mathbb{Z}[x,x^{-1},y] $ becomes a Hopf algebra with comultiplication encoding the group-multiplication. Because

$\begin{bmatrix} 1 & b_1 \\ 0 & a_1 \end{bmatrix} \begin{bmatrix} 1 & b_2 \\ 0 & a_2 \end{bmatrix} = \begin{bmatrix} 1 & 1 \times b_2 + b_1 \times a_2 \\ 0 & a_1 \times a_2 \end{bmatrix} $

we have $\Delta(x) = x \otimes x $ and $\Delta(y) = 1 \otimes y + y \otimes x $, or $x $ is a group-like element whereas $y $ is a skew-primitive. If $R \subset \mathbb{R} $ is a subring of the real numbers, we denote by $P_R^+ $ the subgroup of $P_R $ consisting of all matrices with $a > 0 $. For example,

$\Gamma_0 = P_{\mathbb{Z}}^+ = { \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}~|~n \in \mathbb{Z} } $

which is a subgroup of $\Gamma = P_{\mathbb{Q}}^+ $ and our first job is to describe the cosets.

The left cosets $\Gamma / \Gamma_0 $ are the subsets $\gamma \Gamma_0 $ with $\gamma \in \Gamma $. But,

$\begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & b+n \\ 0 & a \end{bmatrix} $

so if we represent the matrix $\gamma = \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} $ by the point $~(a,b) $ in the right halfplane, then for a given positive rational number $a $ the different cosets are represented by all $b \in [0,1) \cap \mathbb{Q} = \mathbb{Q}/\mathbb{Z} $. Hence, the left cosets are all the rational points in the region between the red and green horizontal lines. For fixed $a $ the cosets correspond to the rational points in the green interval (such as over $\frac{2}{3} $ in the picture on the left.

Similarly, the right cosets $\Gamma_0 \backslash \Gamma $ are the subsets $\Gamma_0 \gamma $ and as

$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} = \begin{bmatrix} 1 & b+na \\ 0 & a \end{bmatrix} $

we see similarly that the different cosets are precisely the rational points in the region between the lower red horizontal and the blue diagonal line. So, for fixed $a $ they correspond to rational points in the blue interval (such as over $\frac{3}{2} $) $[0,a) \cap \mathbb{Q} $. But now, let us look at the double coset space $\Gamma_0 \backslash \Gamma / \Gamma_0 $. That is, we want to study the orbits of the action of $\Gamma_0 $, acting on the right, on the left-cosets $\Gamma / \Gamma_0 $, or equivalently, of the action of $\Gamma_0 $ acting on the left on the right-cosets $\Gamma_0 \backslash \Gamma $. The crucial observation to make is that these actions have finite orbits, or equivalently, that $\Gamma_0 $ is an almost normal subgroup of $\Gamma $ meaning that $\Gamma_0 \cap \gamma \Gamma_0 \gamma^{-1} $ has finite index in $\Gamma_0 $ for all $\gamma \in \Gamma $. This follows from

$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} \begin{bmatrix} 1 & m \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & b+m+an \\ 0 & a \end{bmatrix} $

and if $n $ varies then $an $ takes only finitely many values modulo $\mathbb{Z} $ and their number depends only on the denominator of $a $. In the picture above, the blue dots lying on the line over $\frac{2}{3} $ represent the double coset

$\Gamma_0 \begin{bmatrix} 1 & \frac{2}{3} \\ 0 & \frac{2}{3} \end{bmatrix} $ and we see that these dots split the left-cosets with fixed value $a=\frac{2}{3} $ (that is, the green line-segment) into three chunks (3 being the denominator of a) and split the right-cosets (the line-segment under the blue diagonal) into two subsegments (2 being the numerator of a). Similarly, the blue dots on the line over $\frac{3}{2} $ divide the left-cosets in two parts and the right cosets into three parts.

This shows that the $\Gamma_0 $-orbits of the right action on the left cosets $\Gamma/\Gamma_0 $ for each matrix $\gamma \in \Gamma $ with $a=\frac{2}{3} $ consist of exactly three points, and we denote this by writing $L(\gamma) = 3 $. Similarly, all $\Gamma_0 $-orbits of the left action on the right cosets $\Gamma_0 \backslash \Gamma $ with this value of a consist of two points, and we write this as $R(\gamma) = 2 $.

For example, on the above picture, the black dots on the line over $\frac{2}{3} $ give the matrices in the double coset of the matrix

$\gamma = \begin{bmatrix} 1 & \frac{1}{7} \\ 0 & \frac{2}{3} \end{bmatrix} $

and the gray dots on the line over $\frac{3}{2} $ determine the elements of the double coset of

$\gamma^{-1} = \begin{bmatrix} 1 & -\frac{3}{14} \\ 0 & \frac{3}{2} \end{bmatrix} $

and one notices (in general) that $L(\gamma) = R(\gamma^{-1}) $. But then, the double cosets with $a=\frac{2}{3} $ are represented by the rational b’s in the interval $[0,\frac{1}{3}) $ and those with $a=\frac{3}{2} $ by the rational b’s in the interval $\frac{1}{2} $. In general, the double cosets of matrices with fixed $a = \frac{r}{s} $ with $~(r,s)=1 $ are the rational points in the line-segment over $a $ with $b \in [0,\frac{1}{s}) $.

That is, the Bost-Connes double coset space $\Gamma_0 \backslash \Gamma / \Gamma_0 $ are the rational points in a horrible fractal comb. Below we have drawn only the part of the dyadic values, that is when $a = \frac{r}{2^t} $ in the unit inverval

and of course we have to super-impose on it similar pictures for rationals with other powers as their denominators. Fortunately, NCG excels in describing such fractal beasts…

UPDATE : here is a slightly beter picture of the coset space, drawing the part over all rational numbers contained in the 15-th Farey sequence. The blue segments of length one are at 1,2,3,…

Majority offers security…

Published January 15, 2008 by lievenlb

What better way to start a series on Web 2.0 & Mathematics than to reflect on the potential dangers of majority-approved sites, such as Google, Wikipedia and META-sites. Bee has written a great essay The Spirits That We Called

Now you can tell me everyone of us should be rational, we should always check sources, doubt unverified reports even if repeated several times. We shouldn’t believe what we read without questioning it. We should seek accuracy and not easy entertainment. We should, we should, we should. But face it, many people don’t. Because they just don’t have the time, or are not interested enough, and the most commonly used criteria in this case is to follow the masses. Read what others read (the posts with the most comments?) go where many people link to, talk what others talk about, pay attention to what many people consider relevant. Majority offers security, Wikipedia is trustworthy, Google has proved useful.

To wet your appetite to take a break, and start reading the full essay (11 printed pages available from this pdf link), her opening paragraph, in which she outlines possible consequences of tempering with social bookmark sites.

MARCH 13th 2008: Until Wednesday, the Presidential candidate [insert name here] scored high in the polls. Then a Google search for his name showed up as first hit a report on an alleged child abuse committed by the candidate, published by Mary S. (name changed) on her personal website. The story was backed up by the following highly ranked hits that indicated two similar events during his youth, though reliable sources were missing. Within less than one hour, the reports were echoed on thousands of weblogs, appeared on digg and reddit, the original websites received 200,000 hits within the first 6 hours, until the server crashed down. Immediate press releases by the candidate’s PR groups did not appear on the Google listing, and could only be accessed by secondary links. It took until the next day that printed newspapers could attempt to clarify the situation.

Another, equally interesting essay, to which Bee points is Digital Maoism by Jaron Lanier,

The Wikipedia is far from being the only online fetish site for foolish collectivism. There’s a frantic race taking place online to become the most “Meta” site, to be the highest level aggregator, subsuming the identity of all other sites.
The race began innocently enough with the notion of creating directories of online destinations, such as the early incarnations of Yahoo. Then came AltaVista, where one could search using an inverted database of the content of the whole Web. Then came Google, which added page rank algorithms. Then came the blogs, which varied greatly in terms of quality and importance. This lead to Meta-blogs such as Boing Boing, run by identified humans, which served to aggregate blogs. In all of these formulations, real people were still in charge. An individual or individuals were presenting a personality and taking responsibility.
…
In the last year or two the trend has been to remove the scent of people, so as to come as close as possible to simulating the appearance of content emerging out of the Web as if it were speaking to us as a supernatural oracle. This is where the use of the Internet crosses the line into delusion.
…
In March, Kelly reviewed a variety of “Consensus Web filters” such as “Digg” and “Reddit” that assemble material every day from all the myriad of other aggregating sites. Such sites intend to be more Meta than the sites they aggregate. There is no person taking responsibility for what appears on them, only an algorithm. The hope seems to be that the most Meta site will become the mother of all bottlenecks and receive infinite funding.

Now, please print out these two essays, turn off your computer for an hour, and read them! Perhaps they change your opinion as to whether or not getting involved into some of these bookmark sites. Being aware of their potential danger is one (important) thing, neglecting them altogether has drawbacks too.

You might want to repeat the experiment I performed last night : type in your favorite technical term (mine was ‘noncommutative’, when this failed to return a hit, I tried ‘geometry’) into bookmark sites such as StumbleUpon, Digg, MyBlogLog, Reddit, blinklist, Magnolia and a dozen other similar ones.

You will discover that there is hardly any mathematics of value to be found there. As more people are using such sites in search of information, an inevitable consequence is that mathematics will become even more marginal, unless we take some action.

In my experiment, there was one noteworthy exception (( delicious was another ok-site )) : CiteUlike which has 427 articles tagged noncommutative, perhaps a result of the action I started 2 years ago. So, there is still hope!

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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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Quiver-superpotentials

Published January 14, 2008 by lievenlb

It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma $, or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of $\mathbb{H} / \Lambda $ where $\mathbb{H} $ is the hyperbolic upper half-plane. This quiver is independent of the chosen embedding of the dessin in the Dedeking tessellation. (For more on these terms and constructions, please consult the series Modular subgroups and Dessins d’enfants).

Why are quivers useful? To start, any quiver $Q $ defines a noncommutative algebra, the path algebra $\mathbb{C} Q $, which has as a $\mathbb{C} $-basis all oriented paths in the quiver and multiplication is induced by concatenation of paths (when possible, or zero otherwise). Usually, it is quite hard to make actual computations in noncommutative algebras, but in the case of path algebras you can just see what happens.

Moreover, we can also see the finite dimensional representations of this algebra $\mathbb{C} Q $. Up to isomorphism they are all of the following form : at each vertex $v_i $ of the quiver one places a finite dimensional vectorspace $\mathbb{C}^{d_i} $ and any arrow in the quiver
[tex]\xymatrix{\vtx{v_i} \ar[r]^a & \vtx{v_j}}[/tex] determines a linear map between these vertex spaces, that is, to $a $ corresponds a matrix in $M_{d_j \times d_i}(\mathbb{C}) $. These matrices determine how the paths of length one act on the representation, longer paths act via multiplcation of matrices along the oriented path.

A necklace in the quiver is a closed oriented path in the quiver up to cyclic permutation of the arrows making up the cycle. That is, we are free to choose the start (and end) point of the cycle. For example, in the one-cycle quiver

[tex]\xymatrix{\vtx{} \ar[rr]^a & & \vtx{} \ar[ld]^b \\ & \vtx{} \ar[lu]^c &}[/tex]

the basic necklace can be represented as $abc $ or $bca $ or $cab $. How does a necklace act on a representation? Well, the matrix-multiplication of the matrices corresponding to the arrows gives a square matrix in each of the vertices in the cycle. Though the dimensions of this matrix may vary from vertex to vertex, what does not change (and hence is a property of the necklace rather than of the particular choice of cycle) is the trace of this matrix. That is, necklaces give complex-valued functions on representations of $\mathbb{C} Q $ and by a result of Artin and Procesi there are enough of them to distinguish isoclasses of (semi)simple representations! That is, linear combinations a necklaces (aka super-potentials) can be viewed, after taking traces, as complex-valued functions on all representations (similar to character-functions).

In physics, one views these functions as potentials and it then interested in the points (representations) where this function is extremal (minimal) : the vacua. Clearly, this does not make much sense in the complex-case but is relevant when we look at the real-case (where we look at skew-Hermitian matrices rather than all matrices). A motivating example (the Yang-Mills potential) is given in Example 2.3.2 of Victor Ginzburg’s paper Calabi-Yau algebras.

Let $\Phi $ be a super-potential (again, a linear combination of necklaces) then our commutative intuition tells us that extrema correspond to zeroes of all partial differentials $\frac{\partial \Phi}{\partial a} $ where $a $ runs over all coordinates (in our case, the arrows of the quiver). One can make sense of differentials of necklaces (and super-potentials) as follows : the partial differential with respect to an arrow $a $ occurring in a term of $\Phi $ is defined to be the path in the quiver one obtains by removing all 1-occurrences of $a $ in the necklaces (defining $\Phi $) and rearranging terms to get a maximal broken necklace (using the cyclic property of necklaces). An example, for the cyclic quiver above let us take as super-potential $abcabc $ (2 cyclic turns), then for example

$\frac{\partial \Phi}{\partial b} = cabca+cabca = 2 cabca $

(the first term corresponds to the first occurrence of $b $, the second to the second). Okay, but then the vacua-representations will be the representations of the quotient-algebra (which I like to call the vacualgebra)

$\mathcal{U}(Q,\Phi) = \frac{\mathbb{C} Q}{(\partial \Phi/\partial a, \forall a)} $

which in ‘physical relevant settings’ (whatever that means…) turn out to be Calabi-Yau algebras.

But, let us return to the case of subgroups of the modular group and their quivers. Do we have a natural super-potential in this case? Well yes, the quiver encoded a triangulation of the compactification of $\mathbb{H}/\Lambda $ and if we choose an orientation it turns out that all ‘black’ triangles (with respect to the Dedekind tessellation) have their arrow-sides defining a necklace, whereas for the ‘white’ triangles the reverse orientation makes the arrow-sides into a necklace. Hence, it makes sense to look at the cubic superpotential $\Phi $ being the sum over all triangle-sides-necklaces with a +1-coefficient for the black triangles and a -1-coefficient for the white ones. Let’s consider an index three example from a previous post

[tex]\xymatrix{& & \rho \ar[lld]_d \ar[ld]^f \ar[rd]^e & \\
i \ar[rrd]_a & i+1 \ar[rd]^b & & \omega \ar[ld]^c \\
& & 0 \ar[uu]^h \ar@/^/[uu]^g \ar@/_/[uu]_i &}[/tex]

In this case the super-potential coming from the triangulation is

$\Phi = -aid+agd-cge+che-bhf+bif $

and therefore we have a noncommutative algebra $\mathcal{U}(Q,\Phi) $ associated to this index 3 subgroup. Contrary to what I believed at the start of this series, the algebras one obtains in this way from dessins d’enfants are far from being Calabi-Yau (in whatever definition). For example, using a GAP-program written by Raf Bocklandt Ive checked that the growth rate of the above algebra is similar to that of $\mathbb{C}[x] $, so in this case $\mathcal{U}(Q,\Phi) $ can be viewed as a noncommutative curve (with singularities).

However, this is not the case for all such algebras. For example, the vacualgebra associated to the second index three subgroup (whose fundamental domain and quiver were depicted at the end of this post) has growth rate similar to that of $\mathbb{C} \langle x,y \rangle $…

I have an outlandish conjecture about the growth-behavior of all algebras $\mathcal{U}(Q,\Phi) $ coming from dessins d’enfants : the algebra sees what the monodromy representation of the dessin sees of the modular group (or of the third braid group).
I can make this more precise, but perhaps it is wiser to calculate one or two further examples…

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