Thursday, December 6, 2007

Noncommutative Geometry in China

During August 15-30, 2007, a school and workshop on noncommutative geometry took place at the Chern Institute in Nankai University in Tianjin, China. The first two weeks and part of the third week was devoted to a series of mini-courses (5 lectures each) on various aspects of noncommutative geometry. This conference and its success owes a lot to the Chern Institute and to our Chinese friends on the organizing committee. We really had a very good time in Tianjin. A big thank you to all for their hospitality and efforts! Here is a list of mini-courses and speakers:

G. Yu: Cyclic cohomology and Connes-Chern character;
G. Gong: Operator algebras and K-theory;
L. Guo: Rota-Baxter algebra with applications to renormalizations;
A. Thom: L^2-Betti numbers for von Neumann algebras;
T. Natsume: Index theory and noncommutative geometry;
V. Mathai: Geometry of the determinant line bundle in noncommutative geometry;
W. van Suijlekom: Noncommutative geometry and physics;
M. Khalkhali: Noncommutative geometry and the local index formula

The first two mini-courses were meant to warm up the audience of mostly grad students and postdocs for lectures in the next weeks. I asked the mini-course speakers to write a blurb on what they talked and most of them budged. So here are their reports and my apologies for delaying this so long!

Andreas Thom; L^2-invariants and von Neumann algebras:
This mini-course consisted of five lectures around the algebraic theory of l^2-invariants for groups and tracial algebras. After a review of the basic tools of homological algebra and an introduction to Lueck's dimension theory for modules over finite von Neumann algebras, a new and conceptual proof of Gaboriau's theorem about proportionality of l^2-Betti numbers for measure equivalent groups was presented, see (L^2-invariants and rank metric, arXiv:math/OA.0607263). In the third and fourth lecture recent work of Peterson-Thom (Group cocycles and the ring of affiliated operators, arXiv:0708.4327) was presented. Here, the focus was on applications to geometric and combinatorial group theory. Indeed, as was shown in the course, non-vanishing of the first l^2-Betti number of a group implies strong restrictions on the subgroup structure. Several conjectures related to Atiyah's conjecture were discussed in detail. In the fifth lecture, a quick review of the work of Connes-Shlyakhtenko (math.OA/0309343) on l^2-invariants for tracial algebras was given.

Walter van Suijlekom; Noncommutative geometry and physics:
The subject of my lectures in Tianjin was the noncommutative geometry of the Standard Model of high-energy physics. We started in the first two lectures with an overview of the essential parts of the Standard Model in order to recognize it later on when derived in the setting of noncommutative geometry.In the remaining three lectures, we looked at how the so-called Spectral Action Principle allows us to obtain the full (Lagrangian of the) Standard Model from a natural noncommutative geometry given as a real spectral triple. We based our lectures on the book ``Noncommutative Geometry, Quantum Fields. and Motives'' by Connes and Marcolli, appearing at the beginning of 2008.

Toshi Natsume; Index theory and noncommutative geometry:
in this short course historical background was given, and the meaning and basic properties of the index were explained. In fact the Atiyah-Singer Index Theorem holds on all sorts of manifolds, but for this course we focused on elliptic operators on Euclidean space, and proved the index theorem here (still a highly nontrivial result). The course required background in multivariable calculus and linear algebra. An introductory survey of the additional tools (from functional analysis and topology) needed to prove the theorem were given. The Atiyah-Singer Index Theorem opened the door to a new world of interaction between different areas of mathematics, where analytic machinery such as operator algebras can play a significant role in topology and geometry. A crystallization of this idea is noncommutative geometry, currently important particularly in today’s mathematical physics. The course was concluded by explaining some fundamental ideas and remarkable results in noncommutative geometry

Li Guo; Connes-Kreimer algebraic Birkhoff decomposition and applications:
The algebraic Birkhoff decomposition of Connes
and Kreimer is a fundamental result in their Hopf algebra
approach to renormalization of perturbative quantum field
theory. We provide enough background to present this result
and prove it in the context of Rota-Baxter algebras through
Spitzer's identity and Atkinson factorization. We then illustrate
its applications to renormalization in quantum field theory
and in multiple zeta values.

Masoud Khalkhali; Noncommutative geometry and the local index formula:
This was an introduction to the local index formula of Connes and Moscovici (GAFA 95), currently the most general and most elaborate form of an index theorem available in NCG. To understand the theorem, almost all of the key ideas of NCG must be introduced: K-theory and K-homology, cyclic cohomology, index pairing, Connes-Chern character maps, Dixmier trace and spectral zeta functions, and noncommutative residues. As such I found the topic an ideal way of introducing these concepts and tools all with the goal of reaching to one of the summits of NCG.

Here are a few pictures I took in Tianjin

Chairman Mao still tries to lead! (why not?)

A nice painting depicting old friends S. S. Chern and C. N. Yang in the lobby of the Chern Institute. ( I can only imagine what they are talking about......but your `gauge fields' are much the same as my `connections' and your `field strengths' are like my `curvatures'........)

Long Live Noncommutative Geometry!

Wednesday, October 31, 2007


Katia's last post ended with a provocative question motivated by Grothendieck's description in Récoltes et Semailles of the "heart of the heart" of arithmetic geometry, namely the theory of motives. Her question was formulated like this:
--------What is the "heart of the heart" of noncommutative geometry?-------
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...
This opens up completely new ways of handling geometric spaces and our work with Matilde Marcolli and Katia Consani is just one example of potential applications to number theory. It is closely related to the Riemann zeta function and is very close in spirit to Grothendieck's ideas on motives so that it is not out of place in the present discussion of Katia's question.
The story starts by a qualitative distinction between spaces which comes from the classification (by von Neumann) of noncommutative algebras in types I, II and III. The commutative spaces are all of type I. When encoding a space X by an algebra A of (complex valued) functions on X one uses some structure on X to restrict the class of functions (e.g. to smooth functions on a smooth space) and the above distinction between types uses the coarsest possible structure which is the measure theory. The corresponding algebras (called von Neumann algebras) are quite simple to characterize abstractly: they are commutants in Hilbert space of some unitary representation.
Since one can take the direct sum of algebras A and B, one can mix algebras of different types. More precisely any von Neumann algebra decomposes uniquely as an integral of algebras which cannot be decomposed further and are called factors. A factor is a von Neumann algebra whose center is as small as it can be, namely is reduced to the complex numbers. The factors of type I are Morita equivalent to the complex numbers, and thus a type I factor really corresponds to the classical notion of "point" in a space X.
To understand geometrically what factors of type II and III look like, it is useful to describe the (von Neumann) algebra A associated to the leaf space of a foliated manifold: (V,F). An element T of A assigns to each leaf an operator in the Hilbert space of square integrable functions on the leaf, and it makes sense to say that T is bounded, measurable, or zero almost everywhere. The algebraic operations are done leaf per leaf, and the algebra of bounded measurable elements modulo the negligible ones is a von Neumann algebra. The simplest example corresponds to the foliation whose leaf space is the noncommutative torus. It is the foliation of the two torus by the equation "dy= a dx" in flat coordinates. The corresponding von Neumann algebra is a factor when "a" is irrational and this factor is not of type I but of type II. To obtain type III examples one can take any codimension one foliation whose Godbillon-Vey invariant does not vanish. The integrable subbundle F defining a codimension one foliation is the orthogonal of a one form v and integrability gives dv as the wedge product of v by a one form w. The Godbillon-Vey invariant is the integral over V of the wedge product of w by dw when V is compact oriented of dimension three. In essence the form w is the logarithmic derivative of a transverse volume element and the GV invariant is an obstruction to finding a holonomy invariant tranverse volume element ie one which does not change when one moves along a leaf keeping track of the way the nearby leaves are developing.
More generally the factors of type II are those which possess a trace and those of type III are those which are neither of type I nor of type II. In the foliation context, a holonomy invariant tranverse volume element allows one to integrate the ordinary trace of operators and this yields a trace on the von Neumann algebra of the foliation.
Until the work of the Japanese mathematician Minoru Tomita, very few positive results existed on type III factors. The key result of Tomita is that a cyclic and separating vector v for a factor A in a Hilbert space H generates a one parameter group of automorphisms of A by the following recipee: one considers the modulus square S*S of the closable operator S which sends xv to S(xv)=x*v for any x in A, and then raises it to the purely imaginary power "it". Tomita showed that the resulting unitary operator normalizes A and hence defines an automorphism of A. One obtains in this way a one parameter group of automorphisms of A associated to the choice of a cyclic and separating vector v. He also showed that the phase J of the above closable operator S yields an antiisomorphism of A with its commutant A' which coincides with JAJ. In his account of Tomita's work, Takesaki characterized the relation between the state defined by the cyclic and separating vector v and the one parameter group of automorphisms of Tomita as the Kubo-Martin-Schwinger (KMS) condition, which had been formulated in C*-algebraic terms by the physicists Haag, Hugenholtz and Winnink.
The key result of my thesis (in 1972) is that the class modulo inner automorphisms of the Tomita automorphism group is in fact independent of the choice of the (faithful normal) state that is used in its construction. Needless to say it is this uniqueness that allows to define invariants of factors. The simplest is the subgroup T(A) of R which is formed of the periods, namely the set of times t for which the corresponding automorphism is inner. This, together with the spectral invariant S(A), led me to the classification of type III factors into subtypes III_s for s in [0,1] and the reduction from type III to type II and automorphisms done in my thesis except for the case III_1 which was later completed by Takesaki. All of this goes back to the beginning of the seventies and will suffice for this first heart beat. It is only the beginning of a long saga which is far from over hopefully, and whose main theme is this mysterious generation of an intrinsic "time" that emerges from the noncommutativity of a von Neumann algebra. Exactly as manifolds come with a natural "smooth" measure class, a noncommutative space X generally gives rise to a von Neumann algebra A which encodes the natural measure class on X. It is thus a totally new feature of the noncommutative world that the corresponding time evolution is well defined and gives a canonical homomorphism:

where the second line gives the definition of the group of outer automorphisms Out(A) of A as the quotient of the group Aut(A) of automorphisms by the normal subgroup Int(A) of inner automorphisms (which are obtained by conjugating by a unitary element of the algebra A).

Report on the AMS Special Session on Noncommutative Geometry and Arithmetic Geometry

Let us start this report on this meeting in a light way with a picture, featuring the subject of this blog and David Goss...

Connes opened the meeting with a talk on some analogies between two grand challenges in mathematics and physics: On the one hand, the search for a geometric setting in which the methods of Weil's proof of the Riemann hypothesis (RH) for curves over finite fields could be applied to prove the original RH; and on the other hand, the search for a quantum theory of gravity starting from the NCG approach to the standard model of particle physics based on the spectral action principle of Connes and Chamseddine. If my memory from an earlier talk of Connes is correct, these analogies were discovered while Connes and Marcolli were finishing an early draft of their tome "Noncommutative geometry, quantum fields, and motives" (draft here), giving rise to the final part of the book that ties (conjecturally) the two major mathematics and physics strands mentioned above. Let me try to give a (somewhat disjointed) indication of the breadth of the analogies, while leaving the bigger picture completely in the fog.

First, Connes gave an overview of the Tomita-Takesaki theory (the "secret weapon" of operator algebraicists, in the words of Jack Morava), emphasizing its novelty and stark contrast with the commutative case: noncommutative operator algebras -- but not commutative ones! -- come endowed with a non-trivial, canonical (that is, up to alteration by an inner automorphism) time evolution. Surprisingly, the theory of Tomita-Takesaki also provided the correct framework for operator-algebraic quantum statistical mechanics. Now, these are old results from the 70's (and late 60's), but around 1992 in collaboration with C. Rovelli the two points of view were considered together in a novel way: Is there a thermodynamic basis for the origin of time? In particular, what should be the (noncommutative) algebra of observables of a quantum theory of gravity?

Connes also gave a rapid summary of his on-going project with Consani and Marcolli to build a geometric world in characterstic 0 hospitable to the methods of Weil's proof of the Riemann hypothesis for curves over finite fields. Apparently, there are fruitful analogies between the necessary ingredients for quantum gravity (QG) and aspects of the space of Q-lattices, the geometric space underlying the GL(2)-system of Connes-Marcolli and the Bost-Connes system (see the summary of Laca's talk below). For example, the moduli space of Dirac operators on the QG side, being described by a double quotient space of complex algebraic groups, is mirrored on the Q-lattice side by Shimura varieties (certain double quotient spaces of adelic algebraic groups). One will find a condensed dictionary of many more analogies in the last part of the book by Connes and Marcolli. It would be desirable if some knowledgeable reader of this blog could elaborate on this (perhaps even the authors themselves, the huge job of having written 700 or so pages notwithstanding).

Continuing in the bridge-building spirit of the meeting, van Suijlekom gave a talk on his recent work with S. Mahanta on their study of the noncommutative torus from the point of view of noncommutative algebraic geometry. This is a very natural undertaking: for while noncommutative tori have long been studied from a topological and differential perspective, classical tori can also be realized as 1-dimensional complex abelian varieties (a.k.a. elliptic curves) which have rich algebraic and arithmetic structures, so it is natural to try to examine noncommutative tori as noncommutative algebraic varieties of sorts. But whereas in the differential-topological approach pioneered by Connes and Rieffel a noncommutative space is a certain kind of noncommutative topological algebra, in the current algebraic-geometry approach, a noncommutative variety is regarded as a certain type of category. Indeed, from the work of A. Rosenberg, Bondal, and Orlov it is known that smooth (irreducible) projective varieties are characterized up to isomorphism by their bounded derived categories of quasi-coherent sheaves. The work of Mahanta-van Suijlekom is an attempt to connect these two worlds for NC tori. What they have done is to define a category that, roughly speaking, interpolates between the categories reflecting the differential and algebraic nature of the NC tori. Additionally, they've shown that this interpolating category is a Tannakian category equivalent to the category of representations of Z^2.

It would be interesting to see whether the categorical approach to noncommutative tori sheds any light on the conjectured relevance of noncommutative tori to an explicit class field theory for real quadratic fields (in analogy with the theory of complex multiplication, as suggested by Manin), or clarifies what it should mean for a noncommutative torus to be defined over Q or a number field (cf. the recent thesis of J. Plazas).

Laca gave a report on his recent work with N. Larsen and S. Neshveyev. This was an especially pleasing talk to attend as this work finally wraps up an analytic problem that has remained open for more than 10 years, namely the classification of KMS states for the Bost-Connes C*-dynamical system for number fields. Avoiding all details of what the Bost-Connes system is exactly -- an excellent summary is given in the book of Connes and Marcolli -- let me mention only that its most "fabulous" feature is that it admits an action of the abelianized absolute Galois group of Q on its so-called KMS infinity states, and upon evaluation of theses KMS states on a natural rational subalgebra, this Galois action coincides with the usual Galois action on the maximal cyclotomic extension of Q. (KMS-beta states were discussed by Connes in his talk and are surely discussed elsewhere on this blog as well. To describe them quickly, albeit in a rather cryptic manner: KMS-beta states are analogues of infinite-volume limits of Gibbs states in quantum statistical mechanics; beta, in the physical context, is inverse temperature.) A natural problem is to construct C*-dynamical systems with analogous Fabulous Features for general number fields. For the case of imaginary quadratic fields, this was accomplished about three years ago by Connes-Marcolli-Ramachandran. Paugam and one of the blog posters have defined a candidate analogue of Bost-Connes for general number fields, without, however, being able to show that it is fabulous. What Laca and his collaborators have done is overcome a key analytic obstacle towards establishing "fabulousness" of the Bost-Connes system for general number fields: namely, for all beta they have classified the KMS-beta states. The result is essentially the same as for the original Bost-Connes system, though the proof follows the ergodic-theoretic techniques developed by Neshveyev, later enhanced by Laca, Larsen, and Neshveyev to clean up the KMS states classification for the Connes-Marcolli GL(2)-system. To get truly "fabulous" systems in the general number field case capable of manifesting the Galois action, it remains to find an appropriate rational structure for such C*-systems. This is a problem of a different nature, which is not likely to fall without deep arithmetic insight, given that it has implications for Hilbert's 12th problem.

The second day started with two talks by Kreimer and Yeats discussing results obtained in (perturbative) quantum field theories, in particular on quantum electrodynamics. The recursive structures that appear are by now well-known to be captured by the structure of a Hopf algebra. On the analytical side, one can expand the probability amplitudes of interest in physics (such as the vacuum self-energy of the photon) as a series in certain functions gamma_k of the coupling constant. One then writes a recursive relation for the $\gamma_k$ and tries to (numerically) solve a differential equation for the $\gamma_1$. This involved only the computation of the amplitudes of primitive graphs, which was carried out up to fourth order in the loop number. Several vector flow diagrams were presented in the second talk, corresponding to the differential equation. Striking was the difference when moving to 4th loop order, where a separatrix
appeared. Although not yet completely understood, it was observed that the fine structure constant $\alpha = 1/137... lies on this separatrix!

In addition to the talks mentioned above there were also talks on the meeting by Ramachandran on computing Beilinson's ring of correspondences at the generic point of a smooth projective variety over a finite field; by Marcolli on her joint work with Manin on the pseudomeasure formalism for modular symbols (a manifestation of a "modular shadow" in their terminology); by Moscovici on twisted spectral triples (though, unfortunately, there wasn't enough time for him to go deeper into applications to the GL(2)-system of Connes-Marcolli-Moscovici); by Goss on Hecke operators and distributions (in the sense of probability theory) in characteristic p, and some work of Boeckle; and by Zhao on improving the Deligne-Goncharov upper bounds for the dimension of spaces of multiple zeta values (of a given weight). On the second day there were additional talks by Retakh on a construction of Lie algebras and Lie groups over noncommutative rings; by Gangl on Polygons and mixed Tate motives (with Brown and Levin); and finally Zhang on differential renormalization for multiple zeta values (joint with Guo).

Eugene Ha
Walter van Suijlekom

Thursday, September 20, 2007

Two Tracts by Galois

While updating my post on Hilbert, thanks to Manfred Karbe's e-mails I came across these two very interesting, powerful and engaging pieces on abstraction and instruction in mathematics by Galois. I put these texts here just in case the French website where they are taken from becomes unavailable sometime. I think the ultimate reference for Galois' writings is the book `Écrits et Mémoires Mathématiques d'Évariste Galois' (Gauthier-Villars, Paris 1962). It contains all of Galois' writings including his work on abelian integrals. The second tract appears in pages 21-25 in this book. The first tract, only partly reproduced here, appears in full in pages 3-11 of this book.

`` Cecy est un livre de bonne foy. Montagne.

.........................Si avec aussi peu de chances d'être compris, je publie, malgré tout, le fruit de mes veilles, c'est afin de prendre date pour mes recherches, c'est afin que les amis que j'ai formés dans le monde avant qu'on m'enterrât sous les verrous, sachent que je suis bien en vie, c'est peut-être aussi dans l'espérance que ces recherches pourront tomber entre les mains de personnes à qui une morgue stupide n'en interdira pas la lecture, et les diriger dans la nouvelle voie que doit, selon moi, suivre l'analyse dans ses branches les plus hautes. Il faut bien savoir que je ne parle ici que d'analyse pure; mes assertions transportées aux applications les plus directes des mathématiques deviendraient paradoxales.

Les longs calculs algébriques ont d'abord été peu nécessaires au progrès des Mathématiques, les théorèmes fort simples gagnaient à peine à être traduits dans la langue de l'analyse. Ce n'est guère que depuis Euler que cette langue plus brève est devenue indispensable à la nouvelle extension que ce grand géomètre a donnée à la science. Depuis Euler les calculs sont devenus de plus en plus nécessaires, mais de plus en plus difficiles à mesure qu'ils s'appliquaient à des objets de science plus avancés. Dès le commencement de ce siècle, l'algorithme avait atteint un degré de complication tel que tout progrès était devenu impossible par ce moyen, sans l'élégance que les géomètres modernes ont su imprimer à leurs recherches, et au moyen de laquelle l'esprit saisit promptement et d'un seul coup un grand nombre d'opérations.

Il est évident que l'élégance si vantée et à si juste titre, n'a pas d'autre but.

Du fait bien constaté que les efforts des géomètres les plus avancés ont pour objet l'élégance, on peut donc conclure avec certitude qu'il devient de plus en plus nécessaire d'embrasser plusieurs opérations à la fois, parce que l'esprit n'a plus le temps de s'arrêter aux détails.

Or je crois que les simplifications produites par l'élégance des calculs, (simplifications intellectuelles, s'entend; de matérielles il n'y en a pas) ont leurs limites; je crois que le moment arrivera où les transformations algébriques prévues par les spéculations des analystes ne trouveront plus ni le temps ni la place de se produire; à tel point qu'il faudra se contenter de les avoir prévues. Je ne veux pas dire qu'il n'y a plus rien de nouveau pour l'analyse sans ce secours: mais je crois qu'un jour sans cela tout serait épuisé.

Sauter à pieds joints sur ces calculs; grouper les opérations, les classer suivant leurs difficultés et non suivant leurs formes; telle est, suivant moi, la mission des géomètres futurs; telle est la voie où je suis entré dans cet ouvrage.

Il ne faut pas confondre l'opinion que j'émets ici, avec l'affectation que certaines personnes ont d'éviter en apparence toute espèce de calcul, en traduisant par des phrases fort longues ce qui s'exprime très brièvement par l'algèbre, et ajoutant ainsi á la longueur des opérations, les longueurs d'un langage qui n'est pas fait pour les exprimer. Ces personnes-là sont en arrière de cent ans.

Ici rien de semblable; ici on fait l'analyse de l'analyse: ici les calculs les plus élevés exécutés jusqu'à présent sont considérés comme des cas particuliers, qu'il a été utile, indispensable de traiter, mais qu'il serait funeste de ne pas abandonner pour des recherches plus larges. Il sera temps d'effectuer des calculs prévus par cette haute analyse et classés suivant leurs difficultés, mais non spécifiés dans leur forme, quand la spécialité d'une question les réclamera.

La thèse générale que j'avance ne pourra être bien comprise que quand on lira attentivement mon ouvrage qui en est une application: non que ce point de vue théorique ait précédé l'application; mais je me suis demandé, mon livre terminé, ce qui le rendrait si étrange à la plupart des lecteurs, et rentrant en moi-même, j'ai cru observer cette tendance de mon esprit à éviter les calculs dans les sujets que je traitais, et qui plus est, j'ai reconnu une difficulté insurmontable à qui voudrait les effectuer généralement dans les matières que j'ai traitées.

On doit prévoir que, traitant des sujets aussi nouveaux, hasardé dans une voie aussi insolite, bien souvent des difficultés se sont présentées que je n'ai pu vaincre. Aussi dans ces deux mémoires et surtout dans le second qui est plus récent, trouvera-t-on souvent la formule "je ne sais pas"). La classe des lecteurs dont j'ai parlé au commencement ne manquera pas d'y trouver à rire. C'est que malheureusement on ne se doute pas que le livre le plus précieux du plus savant serait celui où il dirait tout ce qu'il ne sait pas, c'est qu'on ne se doute pas qu'un auteur ne nuit jamais tant à ses lecteurs que quand il dissimule une difficulté. Quand la concurrence c'est-à-dire l'égoïsme ne règnera plus dans les sciences, quand on s'associera pour étudier, au lieu d'envoyer aux académies des paquets cachetés, on s'empressera de publier ses moindres observations pour peu qu'elles soient nouvelles, et on ajoutera: "je ne sais pas le reste".

De Ste Pélagie, décembre 1831,


The second tract by Galois:

``Sur l'Enseignement des Sciences, des Professeurs, des Ouvrages, des Examinateurs

Je vous serais obligé, si vous voulez bien accueillir les réflexions suivantes, relatives à l'étude des mathématiques dans les collèges de Paris.

D'abord dans les sciences, les opinions ne comptent pour rien; les places ne sauraient être la récompense de telle ou telle manière de voir en politique ou en religion. Je m'informe si un professeur est bon ou mauvais, et je m'inquiète fort peu de sa façon de penser dans des matières étrangères à ses études scientifiques. Ce n'était donc pas sans douleur et indignation que, sous le gouvernement de la restauration, on voyait les places devenir la proie des plus offrants en fait d'idées monarchiques et religieuses. Cet état de choses n'est pas changé; la médiocrité, qui fait preuve de sa répugnance pour le nouvel ordre de choses, est encore privilégiée; et cependant les opinions ne devraient pas être mises en ligne de compte, lorsqu'il s'agit d'apprécier le mérite scientifique des individus.

Commençons par les collèges; là les élèves de mathématiques se destinent pour la plupart à l'école polytechnique; que fait-on pour les mettre en état d'atteindre ce but ? Cherche-t-on à leur faire concevoir le véritable esprit de la science par l'exposé des méthodes les plus simples ? Fait-on en sorte que le raisonnement devienne pour eux une seconde mémoire ? N'y aura-t-il pas au contraire quelque ressemblance entre la manière dont ils APPRENNENT les mathématiques et la manière dont ils APPRENNENT les leçons de français et de latin ? Jadis un élève aurait appris d'un professeur tout ce qui lui est utile de savoir; maintenant il faut le supplément de un, de deux répétiteurs pour préparer un candidat à l'école polytechnique.

Jusques à quand les pauvres jeunes gens seront-ils obligés d'écouter ou de répéter toute la journée ? Quand leur laissera-t-on du temps pour méditer sur cet amas de connaissances, pour coordonner cette foule de propositions sans suite, de calculs sans liaison ? N'y aurait-il pas quelque avantage à exiger des élèves les mêmes méthodes, les mêmes calculs, les mêmes formes de raisonnement, s'ils étaient à la fois les plus simples et les plus féconds ? Mais non, on enseigne minutieusement des théories tronquées et chargées de réflexions inutiles, tandis qu'on omet les propositions les plus simples et les plus brillantes de l'algèbre; au lieu de cela, on démontre à grands frais de calculs et de raisonnements toujours longs, quelquefois faux, des corollaires dont la démonstration se fait d'elle-même.

D'où vient le mal ? Assurément ce n'est pas des professeurs des collèges; ils montrent tous un zèle fort louable; ils sont les premiers à gémir de ce qu'on ait fait de l'enseignement des mathématiques un véritable métier. La cause du mal, c'est aux libraires de MM. les examinateurs qu'il faut la demander. Les libraires veulent de gros volumes: plus il y a de choses dans les ouvrages des examinateurs, plus ils sont certains d'une vente fructueuse; voilà pourquoi nous voyons apparaître chaque année ces volumineuses compilations où l'on voit les travaux défigurés des grands maîtres à côté des essais de l'écolier.

D'un autre côté, pourquoi les examinateurs ne posent-ils les questions aux candidats que d'une manière entortillée ? Il semblerait qu'ils craignissent d'être compris de ceux qu'ils interrogent; d'où vient cette malheureuse habitude de compliquer les questions de difficultés artificielles ? Croit-on donc la science trop facile ? Aussi qu'arrive-t-il ? L'élève est moins occupé de s'instruire que de passer son examen. Il lui faut sur chaque théorie une RÉPÉTITION de chacun des quatre examinateurs; il doit apprendre les methodes qu'ils affectionnent, et savoir à l'avance, pour chaque question et chaque examinateur, quelles doivent être ses réponses et même son maintien. Aussi il est vrai de dire qu'on a fondé depuis quelques années une science nouvelle qui va grandissant chaque jour, et qui consiste dans la connaissance des dégoûts et des préférences scientifiques, des manies et de l'humeur de MM. les examinateurs.

Etes-vous assez heureux pour sortir vainqueur de l'épreuve ? Etes-vous enfin désigné comme l'un des deux cents géomètres à qui on porte les armes dans Paris ? Vous croyez être au bout: vous vous trompez, c'est ce que je vous ferai voir dans une prochaine lettre.

E. G. "


Cette lettre d'Evariste Galois, fut publiée dans Gazette des Ecoles: Journal de l'Instruction Publique, de l'Université, des Séminaires, numéro 110, 2e année, Janvier 1831.
Aucune deuxième lettre n'a été publiée.

Tuesday, September 18, 2007

Les motifs - ou le coeur dans le coeur

It is with this fascinating title that A. Grothendieck presents in Recoltes et Semailles (cfr. Promenade à travers une oeuvre ou l'Enfant et la Mère) the subject of motives: the deepest of the twelve research themes around which he developed his "long-run" research program that literally revolutionized the field of algebraic geometry in the decade 1958-68. Motives were envisaged as the "heart of the heart" of the new geometry (arithmetic geometry) that Grothendieck invented following a scientific strategy based on the introduction of a series of new concepts organized on a progressive level of generality: starting with schemes, topos and sites then continuing with the yoga of motives and motivic Galois groups and finally introducing anabelian algebraic geometry and Galois-Teichmuller theory.
If the notions of scheme and topos were the two crucial ideas which constituted the original driving force in the development of this new geometry -- Grothendieck was evidently fascinated by the concepts of geometric point, space and symmetry -- it is only with the notion of a motive that one eventually captures the deepest structure, the heart of the profound identity between geometry and arithmetic.

Grothendienck wrote very little about motives. The foundations are documented in his unpublished manuscript "Motifs" and were discussed on a seminar at the Institut des Hautes Études Scientifiques, in 1967. We know, by reading Recoltes et Semailles, that he started thinking about motives in 1963-64. J.P. Serre has included in his paper "Motifs" an extract from a letter that Grothendieck wrote to him in August 1964 in which he talks (rather vaguely, in fact) of the notions of motive, fiber functor, motivic Galois group and weights.
Motives were introduced with the ultimate goal to supply an intrinsic explanation for the analogies occurring among the various cohomological theories for algebraic varieties: they were expected to play the role of a universal cohomological theory (the motivic cohomology) and also to furnish a linearization of the theory of algebraic varieties, by eventually providing (this was Grothendieck's viewpoint) the correct framework for a successful attack to the Weil's Conjectures on the zeta-function of an algebraic variety over a finite field.

Unlike in the framework of algebraic topology where the standard cohomological functor is uniquely characterized by the Eilenberg-Steenrod axioms in terms of the normalization associated to the value of the functor on a point, in algebraic geometry there is no suitable cohomological theory with integers coefficients, for varieties defined over a field k, unless one provides an embedding of k into the complex numbers. In fact, by means of such mapping one can form the topological space of the complex points of the original algebraic variety and finally compute the Betti (singular) cohomology. This construction however, does in general depend upon the choice of the embedding of k in the field of complex numbers. Moreover, Hodge cohomology, algebraic de-Rham cohomology, étale l-adic cohomology furnish several examples of different cohomology functors which can be simultaneously associated to a given algebraic variety, each of which supplying a relevant information on the topological space.

Grothendieck theorized that this plethora of different cohomological data should be somewhat encoded systematically within a unique and more general theory of cohomological nature that acts as an internal "liaison" between algebraic geometry and the collection of available cohomological theories. This is the idea of the "motif", namely the common reason behind this multitude of cohomological invariants which governs and controls systematically all the cohomological apparatus pertaining to an algebraic variety or more in general to ascheme.
The original construction of a category M of (pure) motives over a field k starts with two preliminary considerations. The first consideration is that M should be the target of a natural contravariant functor connecting the category C of smooth, projective algebraic varieties over a field k to M. Such functor should map an object X in C to its associated motive M(X). The second consideration is that this functor should, by construction, factor through any particular cohomological theory.
Now, keeping in mind this goal, one thinks about the axiomatizing process of a cohomological theory in algebraic geometry. This is done by introducing a contravariant functor X -> H(X) from C to a graded abelian category, where the sets of morphisms between its objects form K-vector spaces (K is a field of characteristic zero, that for simplicity, I fix here equal to the rationals). One also would like that any correspondence V--> W (an algebraic cycle in the cartesian product VxW that can be view as the graph of a multi-valued algebraic mapping) induces contravariantly, a mapping on cohomology and that the target category is suitably defined so that it contains among its objects any "Weil cohomological theory", namely a cohomology which satisfies among other axioms Poincaré duality and Künneth formula.
This preliminary disquisition helps one in formalizing the construction of the category of motives by following a three-steps procedure. One wishes to enlarges the category C in a precise way with the hope to produce also an abelian category. The three steps are shortly resumed as follows.
(1) One moves from C to a category with the same objects but where the sets of morphisms are the equivalence classes of rational correspondences. Here, the natural choice of the equivalence relation is the numerical equivalence relation as it is the coarsest one among the possible relations between algebraic cycles which can be seen to induce, via the cohomological axioms of any Weil cohomological theory, well-defined homomorphisms in cohomology.
(2) One enlarges the collection of objects of the category defined in (1), by formally adding kernels and images of projectors. This step is technically referred to as the "pseudo-abelian envelope" of the category defined in (1) and it is motivated by the expectation to define an abelian category of motives in which for instance, the Künneth formula can be applied.
(3) Finally, one considers the opposite of the category defined in (2).
Now, after having diligently applied all this abstract machinery, one would like to see a fruitful application of these ideas, in the form, for instance, of the proof of a major conjecture. However, one also perceives quite soon that a successful application of the yoga of motives is subordinated to a thorough knowledge of the theory of algebraic cycles, since the construction of the category M is centered on the idea of enlarging the sets of morphisms by implementing the notion of correspondence. It is for this reason that the Standard Conjectures (cohomological criteria for the existence of interesting algebraic cycles) were associated, since the beginning, to the theory of motives as they seem to play the "conditio sine qua non" a theory of motives has a concrete and successful application.
However, in order to put the Standard Conjectures in the right perspective and to avoid perhaps, an over-estimation of their importance, one should also record that Y. Manin gave in 1968, the first interesting application of these ideas on motives by producing an elegant proof of the Riemann-Weil hypothesis for non-singular three-dimensional projective unirational varieties over a finite field, without appealing to the Standard Conjectures. Moreover, we also know that the Weil's Conjectures have been proved by P. Deligne in 1974 without using neither the theory of motives nor the Standard Conjectures.
Almost forty years have passed since these ideas were informally discussed in the "Grothendieck's circle". An enlarged and in part still conjectural theory of mixed motives has in the meanwhile proved its usefulness in explaining conceptually, some intriguing phenomena arising in several areas of pure mathematics, such as Hodge theory, K-theory, algebraic cycles, polylogarithms, L-functions, Galois representations etc.
Very recently, some new applications of the theory of motives to number-theory and quantum field theory have been found or are about to be developed, with the support of techniques supplied by noncommutative geometry and the theory of operator algebras.
In number-theory, a conceptual understanding of the interpretation proposed by A. Connes of the Weil explicit formulae as a Lefschetz trace formula over the noncommutative space of adèle classes, requires the introduction of a generalized category of motives which is inclusive of spaces which are highly singular from a classical viewpoint. Several questions arise already when one considers special types of zero-dimensional noncommutative spaces, such as the space underlying the quantum statistical dynamical system defined by J.B. Bost and Connes in their paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking" (Selecta Math. (3) 1995). This space is a simplified version of the adèle classes and it encodes in its group of symmetries, the arithmetic of the maximal abelian extension of the rationals.
A new theory of endomotives (algebraic and analytic) has been recently developed in "Noncommutative geometry and motives: the thermodynamics of endomotives" (to appear in Advances in Mathematics). The objects of the category of endomotives are noncommutative spaces described by semigroup actions on projective limits of Artin motives (these are among the easiest examples of pure motives, as they are associated to zero-dimensional algebraic varieties). The morphisms in this new category generalize the notion of (algebraic) correspondences and are defined by means of étale groupoids to account for the presence of the semigroup actions.
An open and interesting problem is connected to the definition of a higher dimensional theory of noncommutative motives and in particular the set-up of a theory of noncommutative elliptic motives and modular forms.
A suitable generalization of the yoga of motives to noncommutative geometry has already produced some interesting results in the form, for example, of an analog in characteristic zero of the action of the Weil group on the étale cohomology of an algebraic variety.
It seems quite exciting to pursue these ideas further: the hope is that the motivic techniques, once suitably transferred in the framework of noncommutative geometry may supply useful tools and produce even more substantial applications than those obtained in the classical commutative context.

What is the "heart of the heart" of noncommutative geometry?

Sunday, September 16, 2007

New books on noncommutative geometry

This seems to be a banner year for books on noncommutative geometry. Leading the pack is ``Noncommutative Geometry, Quantum Fields. and Motives" by Alain and Matilde which will be published by the American Mathematical Society in January 2008. Earlier this year we introduced two other books.

Now two other new books that just appeared in the market. Topological and Bivariant K-Theory by Joachim Cuntz, Ralf Meyer, and Jonathan Rosenberg has just been published by Birkhaeuser. In publisher's introduction we read:
``Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.
We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem. "

The second book is Local and Analytic Cyclic Homology by Ralf Meyer and is published by the European Mathematical Society Publication House. Here is an excerpt by the publisher describing the book: ``Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book the author develops and compares these theories, emphasising their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to K-theory, and the Chern–Connes character for K-theory and K-homology"

A note to our readers and authors: if you know of any other books on the subject that is published this year, or will be published soon, please let one of us know. I know of at least one, but we should wait a bit!

Wednesday, September 12, 2007

Wir müssen wissen, wir werden wissen!

With the above words (translation: we must know, we will know), Hilbert ended his 1930 address in Königsberg, at the Congress of the Association of German Natural Scientists and Medical Doctors. A four minute excerpt was broadcast by radio, and is available here. This I believe is one of the earliest audio files of a speech by a mathematician of note, available online.
For the original German text and an English translation click here. The above picture was kindly sent by A. Rivero. (See his interesting comments about the place the picture was taken). Many thanks to him!

Now that we are inaugurating a new section in the blog under the label `multimedia' (see Matilde's recent post for an example), I thought time is right to add something I always wanted to share with the readers of this blog. Hilbert's address is not about NCG of course! Some of the relevant underlying philosophical and cultural aspects of the Zeitgeist challenged by Hilbert in his address are briefly discussed here. Notice, however, that Hilbert can be regarded as one of the great grandfathers of NCG and the subject owes a lot to him and his Göttingen school of functional analysis and spectral theory in the years 1900-1912 (Erhard Schmidt, Hermann Weyl, ....; as well as Otto Toeplitz who was not in Göttingen). Hilbert's work was centered around the theory of integral equations and its allied spectral theory as it was mostly motivated by Fredholm's 1900 papers. An immediate dramatic success was Weyl's assymptotic law for the eigenvalues of the Dirichlet problem for Laplacian on bounded domains. This basic result of spectral geometry is one of the foundation stones of NCG as well.

Later abstractions and the development of functional analysis on Hilbert space, by von Neumann and others, led to the theory of operator algebras which is of course one of the sources of NCG.

It is interesting to note that Hilbert's address was just one year before Gödel's incompleteness theorems , which in a way showed that, at least globally, one can not be totally optimistic about the power of formalistic approach in mathematics!

Update (Sept 20, 2007): Manfred Karbe kindly wrote in to share the following interesting information (my sincere thanks to him):
`After reading your post and hearing Hilbert's voice (in the
unmistakable Low Prussian dialect) I luckily found the "Hilbert
Gedenkband" on one of my book shelves, edited by Reidemeister and
published by Springer in 1971..........Attached to this booklet (which is out of print now, no wonder at those times!) was a record which contains the four
minute excerpt that was broadcast by radio. See also
this public appeal to locate the original recording'

Sunday, September 9, 2007

You Tube QFT

For the readers who may be interested in the work Alain and I did on renormalization and the Riemann-Hilbert correspondence, as well as in a general introduction to the Connes-Kreimer theory, I noticed recently that the series of lectures of the minicourse we gave two years ago at Vanderbilt University have now been put on You Tube (or Google Video).

You can find them here:

Lecture 1 (1 h 52 min) by Matilde

Lecture 2 (1 h 7 min) by Alain

Lecture 3 (1 h 10 min) by Alain

Lecture 4 (1 h) by Matilde

Lecture 5 (1 h 30 min) by Matilde

Lecture 6 (1 h 25 min) by Alain

Lecture 7 (1 h 24 min) by Alain

Lecture 8 (1 h 7 min) by Matilde

Wednesday, September 5, 2007

News on K-front

Today the editorial board of the new Journal of K-theory put out a public statement, which we reproduce below:


After several public statements and news articles regarding the Springer journal "K-theory" (KT), and the new "Journal of K-Theory" (JKT) to be distributed by Cambridge University Press (CUP), the mathematical community has become aware of ongoing changes. On behalf of the entire Editorial Board of the new JKT, we want to give as precise a picture of the situation as we can at the moment, especially to the authors.

It is very important to us that the authors should not suffer as a result of the transition.
Those authors who submitted papers to KT before August 2007, regardless of whether the paper has already been accepted or is just awaiting review, have three choices:
1) Choose another journal.
2) Maintain submission with KT for final review if necessary and publication if accepted.
3) Transfer their article to the new JKT.
All authors who have not yet done so should please notify Professor Bak on the one hand, Professors Lueck and Ranicki on the other hand, about their choice, as soon as possible. For those who opt for choice #2, Professors Lueck and Ranicki have promised to take over the remaining editorial duties.
We can guarantee that the authors who choose option (3) will have a smooth transition, with their articles progressing as if there has been no change. We will also do everything we can to help those who choose options (1) and (2). In particular, if the authors instruct us, we will be happy to forward to the journals of their choice the full information regarding the status of their articles.

In 2004, because of growing dissatisfaction with Springer, the editorial board of KT authorized Prof. Anthony Bak, the Editor in Chief, to begin negotiations with other publishers. The editorial board was unhappy with the poor quality of the work done by Springer, for example the huge number of misprints in the published version of the articles, the long delay in publication and the high prices Springer was charging.
The negotiations came to a conclusion in 2007. A new journal, entitled "Journal of K-theory" (JKT) will commence publication in late 2007. It will be printed by Cambridge University Press. Papers will appear earlier online, as 'forthcoming articles'.

The title of JKT is currently owned by a private company. This situation is only meant as a temporary solution to restart publication of K-theory articles as soon as possible. It is the Board's intention to create a non-profit academic foundation and to transfer ownership of JKT to this foundation, as soon as possible, but no later than by the end of 2009, a delay justified by many practical considerations.

This shift towards more academic control of journals is not new. We follow here a path opened by Compositio Mathematica, Commentarii Mathematici Helvetici, and others (see for instance the interesting paper of Gerard van der Geer which appeared in the Notices of the AMS in May 2004). We believe that such changes can help keep prices low.

We trust in Prof. Bak's leadership for the launching of JKT and forming, together with the editorial board, the foundation to house the Journal. The statutes of the foundation will provide democratic rules governing the future course and development of the journal, including the election of the managing team.

We hope to have provided a fair picture of the current situation, and we plan to issue another public statement when new developments come up. In case of further questions, please contact any of the signatories.

Let us conclude from a broader perspective: The editorial board is committed to secure the journal's quality and long-term sustainability.

A. Bak
P. Balmer
S. Bloch
G. Carlsson
A. Connes
E. Friedlander
M. Hopkins
B. Kahn
M. Karoubi
G. Kasparov
A. Merkurjev
A. Neeman
T. Porter
J. Rosenberg
A. Suslin
Guoping Tang
B. Totaro
V. Voevodsky
C. Weibel
Guoliang Yu

Monday, August 13, 2007

Harmonic mean

This "post" is mainly an attempt to see if one can manage to use formulas in the blog and discuss some real stuff somehow. The formulas should be really visible, a bit like with transparencies. So as a pretext, I'll start by discussing an issue related to the basic Ansatz:

which gives the operator theoretic line element in terms of the Dirac operator in the general framework of "metric" noncommutative geometry. The kernel of the operator D is finite dimensional and one takes ds to vanish on that kernel. As was already discussed here, the knowledge of D gives back the metric. Moreover the noncommutative integral, in the form of the Dixmier trace, gives back the volume form. Thus the integral of a function f in dimension n is simply given by

where the "cut" integral is the Dixmier trace ie the functional that assigns to an infinitesimal of order one the coefficient of the logarithmic divergency in the series that gives the sum of its eigenvalues.

I will not try to justify the heuristic definition of the line element any further. It is more interesting to put it to the test, to question it, and I will discuss an example of an issue which left me perplex for quite sometime but has a pretty resolution.

The point is to understand what happens when one takes the product of two noncommutative geometries. One gets the following relation for the squares of the corresponding Dirac operators:

where we abuse notations by removing the tensor product by the identity operator that normally goes with each of the operators D_j. Now this relation is quite different from the simple Pythagorean relation of the classical line elements whose square simply add up and it thus raises the question of reconciling the above Ansatz with the simple formula of addition of the squares of the Dirac operators. More generally, one can consider a bunch of NC spaces with Dirac operators D_mu and combine them as follows: One starts with a positive matrix of operators in Hilbert space:

and one extends the above formula giving D^2 for a product of two spaces and forms the following sum:

We make no commutativity hypothesis and even drop the self-adjointness of D_mu which is not needed. We want a formula for the inverse of the square of D ie for:

in terms of the inverse matrix:
which plays a role similar to the g\mu\nu of Riemannian geometry, and of the operators
where the notation with z stresses the fact that we do not even assume self-adjointness of the various D_\mu.

It sounds totally hopeless since one needs a formula for the inverse of a sum of noncommuting operators. Fortunately it turns out that there is a beautiful simple formula that does the job in full generality. It is reminiscent of the definition of distances as an infimum. It is given by:

The infimum is taken over all decompositions of the given vector as a sum:

Note that this formula suffices to determine the operator ds^2 completely, since it gives the value of the corresponding positive quadratic form on any vector in Hilbert space. The proof of the formula is not difficult and can be done by applying the technique of Lagrange multipliers to take care of the above constraint on the free vectors \xi^\mu.

Tuesday, August 7, 2007

The Editorial Board of 'K-Theory' has resigned.

This just in. The editorial board of `K-theory' has resigned and a new journal titled `Journal
of K-theory' has been established. In recent years `K-theory' was published by Springer. `Journal of K-theory' will be published by the Cambridge University Press. The board of editors of the new `Journal of K-theory' has issued an open letter to the mathematics community that we quote in part:

from the Board of Editors
of the Journal of K-theory

Dear fellow mathematicians,

The Editorial Board of 'K-Theory' has resigned. A new journal titled
'Journal of K-theory' has been formed, with essentially the same Board
of Editors............The new journal is to be distributed by Cambridge University Press.
The price is 380 British pounds, which is significantly less than
half that of the old journal. Publication will begin in January 2008.
We ask for your continued support, in particular at the current time.
Your submissions are welcome and may be sent to any of the editors.

Board of Editors
Journal of K-theory''

Friday, August 3, 2007

Paul's Seventies

I am just back from a very nice event around Paul Baum's seventies, which took place in Warsaw last Monday, thanks in particular to Piotr Hajac. I have known Paul since the summer of 1980 when we first met in Kingston. I really had, when I first met him, the impression of meeting l'"Homologie en personne". The more I got to know him through our very long collaboration, the better I enjoyed his clarity of mind and his relentless quest for simplicity and beauty. In many ways he succeeds in doing something very difficult, which Grothendieck advised in "Récoltes et Semailles", namely to keep "une innocence enfantine" in front of mathematics.

The dinner on monday night was comparable in intensity to the memorable one in Martin Walter's place, in Boulder, for Paul's sixties when the team Paul Baum---Raoul Bott forced Martin to search (again and again) his cellar for more bottles of wine to keep up with their drinking ability!.

Raoul Bott died in December 2005. Not long before, Paul went all the way to California to visit him and they talked together for an entire day. This type of faithfulness in friendship and understanding of what really matters, is an attitude towards life which Paul has and which I truly admire.

Wednesday, July 25, 2007


We should perhaps update this blog more often but we find almost no time to do that! Alain is very busy editing, correcting, and finalizing his book, Noncommutative Geometry, Quantum Fields and Motives, with Matilde Marcolli. This is no easy task given the size and scope of the work. BTW, I hope sometime soon we will have a good and extensive review of this book in this blog. I am also busy traveling, going to conferences and giving talks.

Last week I attended the Max Planck Institute conference on Hochschild and cyclic cohomology. The meeting was on Hopf cyclic cohomology and higher homotopy structures on Hochschild and cyclic complexes (the so called `stringy topology'). Somehow both topics are very hot these days and I shall report on some of the talks there later. This is the second in a series of three conferences this summer, all co-organized by Matilde, at MPI devoted to noncommutative geometry and its applications.

Another summer conference is a meeting in Warsaw in honor of Paul Baum's 70 th birthday. Happy birthday Paul!

Tuesday, July 17, 2007

Non Standard stuff

I am not sure I really know how to make use of a "blog" like this one. Recently I had to write a sollicited paper describing the perspective on the structure of space-time obtained from the point of view of noncommutative geometry. At first I thought that I could just be lazy and after the paper was written (it is available here) just use pieces of it to keep this blog alive during the summer vacations. However, when trying to do that, I realized that it was better (partly because of the impractical use of latex in the blog) to first make the paper available and then tell in the blog the additional things one would not "normally" write in a paper (even a non-technical general public paper such as the above). I am not keen on turning the blog into a place for controversies since it is unclear to me that one gains a lot in such discussions. The rule seems to be that, most often, people have prejudices against new stuff mostly because they dont know enough and take the lazy attitude that it is easier to denigrate a theory than to try and appreciate it. I am no exception and have certainly adopted that attitude with respect to supersymmetry or string theory. A debate will usually exhibit the strong opinions of the various sides and it is rare that one witnesses a real change taking place. So much for the "controversy" side. However I do believe that there are some points that can be quite useful to know and which, provided they are presented in a non-polemic manner can help a lot to avoid some pitfalls. I will discuss as an example the two notions of "infinitesimals" that I know and try to explain the relevance of both. This is not a "math paper" but rather an informal discussion.
When I was a student in Ecole Normale about 40 years ago, I fell in love with a new math topic called "nonstandard analysis" which was advocated by A. Robinson. Being a student of Gustave Choquet at that time, I knew a lot about ultrafilters. These maximal filters were (correct me if I am wrong) discovered by H. Cartan during a Bourbaki workshop. At that time Cartan had no name for the new objects but he had found the remarkable efficiency they had in any proof where a compactness and choice arguments were needed. So (this I heard from Cartan) the name he was using was "boum" !!! Of course he knew that it gave a one line proof of the existence of Haar measure (boum...). And also that because of uniqueness of the latter it was in fact proving a rather strong convergence statement on the counting functions that approximate the Haar measure. He wanted to make sure, and wrote in a Compte-Rendu note the full details of a direct geometric argument proving the expected convergence. From ultrafilters to ultraproducts is an easy step. And I got completely bought by ultraproducts when I learnt (around that time) about the Ax-Cochen theorem: the ultraproduct of p-adic fields is isomorphic to the ultraproduct of local function fields with the same residue fields. Thus I started trying to work in that subject and obtained, using a specific class of ultrafilters called "selective", a construction of minimal models in nonstandard analysis. They are obtained as ultraproducts but the ultrafilters used are so special that, for instance, in order to know the element of the ultrapower of a set X, one does not need to care about the labels: the image ultrafilter in X is all that is needed. I wrote a paper explaining how to use ultraproducts and always kept that tool ready for use later on. I used it in an essential manner in my work on the classification of factors. So much for the positive side of the coin. However, quite early on I had tried in vain to implement one of the "selling adds" of nonstandard analysis, namely that it was finally giving the promised land for "infinitesimals". In fact the adds came with a specific example: a purported answer to the naive question "what is the probability "p" that a dart will land at a given point x of the target" in playing a game of darts. This was followed by 1) the simple argument why that positive number "p" was smaller than epsilon for any positive real epsilon 2) one hundred pages of logic 3) the identification of "p" with a "non-standard" number...
At first I attributed my inability to concretely get "p" to my lack of knowledge in logics, but after realizing that the models could be constructed as ultraproducts this excuse no longer applied. At this point I realized that there is some fundamental reason why one will never be able to actually "pin down" this "p" among non-standard numbers: from a non-standard number (non-trivial of course) one canonically deduces a non-measurable character of the infinte product of two element groups (the argument is simpler using a non-standard infinite integer "n", just take the map which to the sequence a_n (of 0 and 1) assigns its value for the index "n"). Now a character of a compact group is either continuous or non-measurable. Thus a non-standard number gives us canonically a non-measurable subset of [0,1]. This is the end of the rope for being "explicit" since (from another side of logics) one knows that it is just impossible to construct explicitely a non-measurable subset of [0,1]!
It took me many years to find a good answer to the above naive question about "p". The answer is explained in details here. It is given by the formalism of quantum mechanics, which as explained in the previous post on "infinitesimal variables" gives a framework where continuous variables can coexist with infinitesimal ones, at the only price of having more subtle algebraic rules where commutativity no longer holds. The new infinistesimals have an "order" (an infinitesimal of order one is a compact operator whose characteristic values \mu_n are a big O of 1/n). The novel point is that they have an integral, which in physics terms is given by the coefficient of the logarithmic divergence of the trace. Thus one obtains a new stage for the "calculus" and it is at the core of noncommutative differential geometry.

In Riemannian geometry the natural datum is the square of the line element, so that when computing the distance d(A,B) between two points one has to minimize the integral from A to B along a continuous path of the square root of g_\mu\nu dx\mu dx\nu. Now it is often true that "taking a square root" in a brutal manner as in the above equation is hiding a deeper level of understanding. In fact this issue of taking the square root led Dirac to his famous analogue of the Schrodinger equation for the electron and the theoretical discovery of the positron. Dirac was looking for a relativistic invariant form of the Schrodinger equation. One basic property of that equation is that it is of first order in the time variable. The Klein-Gordon equation which is the relativistic form of the Laplace equation, is relativistic invariant but is of second order in time. Dirac found away to take the square root of the Klein-Gordon operator using Clifford algebra. In fact (as pointed out to me by Atiyah) Hamilton had already written the magic combination of partial derivatives using his quaternions as coefficients and noted that this gave a square root of the Laplacian. When I was in St. Petersburg for Euler's 300'th, I noticed that Euler could share the credit for quaternions since he had explicitly written their multiplication rule in order to show that the product of two sums of 4 squares is a sum of 4 squares.
So what is the relation between Dirac's square root of the Laplacian and the above issue of taking the square root in the formula for the distance d(A,B). The point is that one can use Dirac's solution and rewrite the same geodesic distance d(A,B) in the following manner: one no longer measures the minimal length of a continuous path but one measures the maximal variation of a function: ie the absolute value of the difference f(A)-f(B). Of course without a restriction on f this would give infinity, but one requires that the commutator [D,f] of f with the Dirac operator is bounded by one. Here we are in our "quantized calculus" stage, so that both the functions on our geometric space as well as the Dirac operator are all concretely represented in the same Hilbert space H. H is the Hilbert space of square integrable spinors and the functions act by pointwise multiplication. The commutator [D,f] is the Clifford mulltiplication by the gradient of f so that when the function f is real, its norm is just the sup norm of the gradient. Then saying that the norm of [D,f] is less than one is the same as asking that f be a Lipschitz function of constant one ie that the absolute value of f(A)-f(B) is less than d(A,B) where the latter is the geodesic distance. For complex valued functions one only gets an inequality, but it suffices to show that the maximum variation of such f gives exactly the geodesic distance: ie we recover the geodesic distance d(A,B) as Sup f(A)-f(B) for norm of [D,f] less than one.
Note that D has the dimension of the inverse of a length, ie of a mass. In fact in the above formula for distances in terms of a supremum the product of "f" by D is dimensionless and "f" has the dimension of a length since f(A) - f(B) is a distance.
Now what is the intuitive meaning of D? Note that the above formula measuring the distance d(A,B) as a supremum is based on the lack of commutativity between D and the coordinates "f" on our space. Thus there should be a tension that prevents D from commuting with the coordinates. This tension is provided by the following key hypothesis "the inverse of D is an infinitesimal".
Indeed we saw in a previous post that variables with continuous range cannot commute with infinitesimals, which gives the needed tension. But there is more, because of the fundamental equation ds = 1/D which gives to the inverse of D the heuristic meaning of the line element. This change of paradigm from the g_\mu\nu to this operator theoretic ds is the exact parallel of the change of the unit of length in the metric system to a spectral paradigm.
Thus one can think of a geometry as a concrete Hilbert space representation not only of the algebra of coordinates on the space X we are interested in, but also of its infinitesimal line element ds. In the usual Riemannian case this representation is moreover irreducible. Thus in many ways this is analogous to thinking of a particle as Wigner taught us, ie as an irreducible representation (of the Poincaré group).

Tuesday, July 10, 2007

A brief history of the metric system

The next step is to understand what is the replacement of the Riemannian paradigm for noncommutative spaces. To prepare for that, and using the excuse of the summer holidays, let me first tell the story of the change of paradigm that already took place in the metric system with the replacement of the concrete "mètre étalon" by a spectral unit of measurement.

The notion of geometry is intimately tied up with the measurement of length. In the real world such measurement depends on the chosen system of units and the story of the most commonly used system--the metric system--illustrates the difficulties attached to reaching some agreement on a physical unit of length which would unify the previous numerous existing choices. As is well known, the United States are one of the few countries that are not using the metric system and this lack of uniformity in the choice of a unit of length became painfully obvious when it entailed the loss of a probe worth 125 million dollars just because two different teams of engineers had used the two different units (the foot and the metric system).

In 1791 the French Academy of Sciences agreed on the definition of the unit of length in the metric system, the "mètre", as being the ten millionth part of the quarter of the meridian of the earth. The idea was to measure the length of the arc of the meridian from Barcelone to Dunkerque while the corresponding angle (approximately 9.5°) was determined using the measurement of latitude from reference stars. In a way this was just a refinement of what Eratosthenes had done in Egypt, 250 years BC, to measure the size of the earth (with a precision of 0.4 %).

Thus in 1792 two expeditions were sent to measure this arc of the meridian, one for the Northern portion was led by Delambre and the other for the southern portion was led by Mechain. Both of them were astronomers who were using a new instrument for measuring angles, invented by Borda, a French physicist. The method they used is the method of triangulation and of concrete measurement of the "base" of one triangle. It took them a long time to perform their measurements and it was a risky enterprize. At the beginning of the revolution, France entered in a war with Spain. Just try to imagine how difficult it is to explain that you are trying to define a universal unit of length when you are arrested at the top of a mountain with very precise optical instruments allowing you to follow all the movements of the troops in the surrounding.
Both Delambre and Mechain were trying to reach the utmost precision in their measurements and an important part of the delay came from the fact that this reached an obsessive level in the case of Mechain. In fact when he measured the latitude of Barcelone he did it from two different close by locations, but found contradictory results which were discordant by 3.5 seconds of arc. Pressed to give his result he chose to hide this discrepancy just to "save the face" which is the wrong attitude for a Scientist. Chased from Spain by the war with France he had no second chance to understand the origin of the discrepancy and had to fiddle a little bit with his results to present them to the International Commission which met in Paris in 1799 to collect the results of Delambre and Mechain and compute the "mètre" from them. Since he was an honest man obsessed by precision, the above discrepancy kept haunting him and he obtained from the Academy to lead another expedition a few years later to triangulate further into Spain. He went and died from malaria in Valencia. After his death, his notebooks were analysed by Delambre who found the discrepancy in the measurements of the latitude of Barcelone but could not explain it. The explanation was found 25 years after the death of Mechain by a young astronomer by the name of Nicollet, who was a student of Laplace. Mechain had done in both of the sites he had chosen in Barcelone (Mont Jouy and Fontana del Oro) a number of measurements of latitude using several reference stars. Then he had simply taken the average of his measurements in each place. Mechain knew very well that refraction distorts the path of light rays which creates an uncertainty when you use reference stars that are close to the horizon. But he considered that the average result would wipe out this problem. What Nicollet did was to ponder the average to eliminate the uncertainty created by refraction and, using the measurements of Mechain, he obtained a remarkable agreement (0.4 seconds ie a few meters) between the latitudes measured from Mont Jouy and Fontana del Oro. In other words Mechain had made no mistake in his measurements and could have understood by pure thought what was wrong in his computation. I recommend the book of Ken Adler for a nice account of the full story of the two expeditions.
In any case in the meantime the International commission had taken the results from the two expeditions and computed the length of the ten millionth part of the quarter of the meridian using them. Moreover a concrete platinum bar with approximately that length was then realized and was taken as the definition of the unit of length in the metric system. With this unit the actual length of the quarter of meridian turns out to be 10 002 290 rather than the aimed for 10 000 000 but this is no longer relevant.
In fact in 1889 the reference became another specific metal bar (of platinum and iridium) called "mètre étalon", which was deposited near Paris in the pavillon de Breteuil. This definition held until 1960.

Already in 1927, at the seventh conference on the metric system, in order to take into account the inevitable natural variations of the concrete called "mètre étalon", the idea emerged to compare it with a reference wave length (the red line of Cadmium).
Around 1960 the reference to the called "mètre étalon" was finally abandoned and a new definition of the unit of length in the metric system (the "mètre) was adopted as 1650763.73 times the wave length of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the Krypton 86Kr.
In 1967 the second was defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of Caesium-133. Finally in 1983 the "mètre" was defined as the distance traveled by light in 1/299792458 second. In fact the speed of light is just a conversion factor and to define the "mètre" one gives it the specific value of c= 299792458 m/s. In other words the "mètre" is defined as a certain fraction 9192631770/299792458~ 30.6633... of the wave length of the radiation coming from the transition between the above hyperfine levels of the Caesium atom.

The advantages of the new standard of length are many. First by not being tied up with any specific location, it is in fact available anywhere where Caesium is. The choice of Caesium as opposed to Helium or Hydrogen which are much more common in the universe is of course still debatable, and it is quite possible that a new standard will soon be adopted involving spectral lines of Hydrogen instead of Caesium. See this paper of Bordé for an update.

While it would be difficult to communicate our standard of length with other extra terrestrial civilizations if they had to make measurements of the earth (such as its size) the spectral definition can easily be encoded in a probe and sent out. In fact spectral patterns provide a perfect "signature" of chemicals, and a universal information available anywhere where these chemicals can be found, so that the wave length of a specific line is a perfectly acceptable unit, while if you start thinking a bit you will find out that we would be unable to just tell where the earth is in the universe... Coordinates ? yes but whith respect to which system? One possibility would be to give the sequence of redshifts to nearby galaxies, and in a more refined manner to nearby stars but it would be quite difficult to be sure that this would single out a definite place.

Tuesday, July 3, 2007

Noncommutative spacetime

As I explained in a previous post, it is only because one drops commutativity that, in the calculus, variables with continuous range can coexist with variables with countable range. In the classical formulation of variables, as maps from a set X to the real numbers, we saw above that discrete variables cannot coexist with continuous variables.
The uniqueness of the separable infinite dimensional Hilbert space cures that problem, and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute.

One way to understand the transition from the commutative to the noncommutative is that in the latter case one needs to care about the ordering of the letters when one is writing.
As an example, use the "commutative rule" to simplify the following cryptic message I received from a friend :"Je suis alençonnais, et non alsacien. Si t'as besoin d'un conseil nana, je t'attends au coin annales. Qui suis-je?"
It is Heisenberg who discovered that such care was needed when dealing with the coordinates on the phase space of microscopic systems.
At the philosophical level there is something quite satisfactory in the variability of the quantum mechanical observables. Usually when pressed to explain what is the cause of the variability in the external world, the answer that comes naturally to the mind is just: the passing of time. But precisely the quantum world provides a more subtle answer since the reduction of the wave packet which happens in any quantum measurement is nothing else but the replacement of a "q-number" by an actual number which is chosen among the elements in its spectrum. Thus there is an intrinsic variability in the quantum world which is so far not reducible to anything classical. The results of observations are intrinsically variable quantities, and this to the point that their values cannot be reproduced from one experiment to the next, but which, when taken altogether, form a q-number.

Heisenberg's discovery shows that the phase-space of microscopic systems is noncommutative inasmuch as the coordinates on that space no longer satisfy the commutative rule of ordinary algebra. This example of the phase space can be regarded as the historic origin of noncommutative geometry. But what about spacetime itself ? We now show why it is a natural step to pass from a commutative spacetime to a noncommutative one.
The full action of gravity coupled with matter admits a huge natural group of symmetries. The group of invariance for the Einstein-Hilbert action is the group of diffeomorphisms of the manifold and the invariance of the action is simply the manifestation of its geometric nature. A diffeomorphism acts by permutations of the points so that points have no absolute meaning.
The full group of invariance of the action of gravity coupled with matter is however richer than the group of diffeomorphisms of the manifold since one needs to include something called ``the group of gauge transformations" which physicists have identified as the symmetry of the matter part.
This is defined as the group of maps from the manifold to some fixed other group, G, called the `gauge group', which as far as we known is: G=U(1).SU(2).SU(3). The group of diffeomorphisms acts on the group of gauge transformations by permutations of the points of the manifold and the full group of symmetries of the action is the semi-direct product of the two groups (in the same way, the Poincaré group which is the invariance group of special relativity, is the semi-direct product of the group of translations by the group of Lorentz transformations). In particular it is not a simple group (a simple group is one which cannot be decomposed into smaller pieces, a bit like a prime number cannot be factorized into a product of smaller numbers) but is a ``composite" and contains a huge normal subgroup.
Now that we know the invariance group of the action, it is natural to try and find a space X whose group of diffeomorphisms is simply that group, so that we could hope to interpret the full action as pure gravity on X. This is the old Kaluza-Klein idea. Unfortunately this search is bound to fail if one looks for an ordinary manifold since by a mathematical result, the connected component of the identity in the group of diffeomorphisms is always a simple group, excluding a semi-direct product structure as that of the above invariance group of the full action of gravity coupled with matter.
But noncommutative spaces of the simplest kind readily give the answer, modulo a few subtle points. To understand what happens note that for ordinary manifolds the algebraic object corresponding to a diffeomorphism is just an automorphism of the algebra of coordinates i.e. a transformation of the coordinates that does not destroy their algebraic relations. When an involutive algebra A is not commutative there is an easy way to construct automorphisms.
One takes a unitary element u of the algebra i.e. such that u u*=u*u=1. Using u one obtains an automorphism called inner, by the formula x -> uxu*.
Note that in the commutative case this formula just gives the identity automorphism (since one could then permute x and u*). Thus this construction is interesting only in the noncommutative case. Moreover the inner automorphisms form a subgroup denoted Int(A) which is always a normal subgroup of the group of automorphisms of A.
In the simplest example, where we take for A the algebra of smooth maps from a manifold M to the algebra of matrices of complex numbers, one shows that the group Int(A) in that case is (locally) isomorphic to the group of gauge transformations i.e. of smooth maps from M to the gauge group G= PSU(n) (quotient of SU(n) by its center). Moreover the relation between inner automorphisms and all automorphisms becomes identical to the exact sequence governing the structure of the above invariance group of the full action of gravity coupled with matter.

It is quite striking that the terminology coming from physics: internal symmetries agrees so well with the mathematical one of inner automorphisms. In the general case only automorphisms that are unitarily implemented in Hilbert space will be relevant but modulo this subtlety one can see at once from the above example the advantage of treating noncommutative spaces on the same footing as the ordinary ones. The next step is to properly define the notion of metric for such spaces and we shall indulge, in the next post, in a short historical description of the evolution of the definition of the ``unit of length" in physics. This will prepare the ground for the introduction to the spectral paradigm of noncommutative geometry.