F_un : day two...

By A. C. and K. Consani

On the second day of the workshop we presented the main results contained in our recent preprint (joint work with Matilde Marcolli) "Fun with F_1". The pdf file is downloadable here. Our main point is to establish a bridge between F_1 and the quantum statistical mechanical system (known as the BC-system) which gives, after passing to the dual system, a spectral realization of the zeros of the Riemann zeta function, as well as a trace formula interpretation of the Riemann-Weil explicit formulas. While a precise algebraic definition of F_1 is still elusive and premature, the section 13 of the paper "Sur les analogues algébriques des groupes semi-simples complexes" of J. Tits gives the construction of the geometries corresponding to the classical Dynkin diagrams starting from the 4 basic rank two cases: A_1 x A_1, A_2, B_2, G_2 which (over F_1) are associated to the geometries of the polygons with 2, 3, 4 and 6 sides. A striking result of J. Tits is that the groups associated to the geometries over F_1 are the Weyl groups. For instance GL_n(F_1) is the symmetric group S_n of permutations of n letters. Thus, in particular the series A_n suggests to view the category of "finite dimensional vector spaces over F_1" as the category of finite sets (with an additional "0" element). Further unpublished work of Kapranov and Smirnov suggests that, similarly to what happens for F_q, one should have for each integer n an extension F_1^n of degree n of F_1 and the category of "finite dimensional vector spaces over F_1^n" should be that of finite sets with a free action of the cyclic group Z/n of order n (still with a "0" added). This lead Soulé to the following equation :

which provides a definition of the ring obtained by "extension of scalars" from F_1 to Z. Now, for q a prime power, the extensions F_q^n can be organized (after choosing an algebraic closure of F_q) as an inductive system with inclusion of F_q^n in F_q^m when n divides m. For such a pair n,m one has similarly a natural inclusion of the cyclic group Z/n in the cyclic group Z/m which relates (by two adjoint functors) the categories of "finite dimensional vector spaces" over F_1^n and F_1^m. Thus one gets

But the algebra on the right hand side is the abelian part of the BC-system which thus arises naturally by extension of scalars from the field with one element.. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions F_1^n, while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for the BC endomotive, which is related to the original Hecke algebra description. Finally, we described the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.

The talk by Henri Moscovici "Spectral triples and Q-lattices" constructs the missing type-III (twisted) spectral triple for the GL(2) analogue of the BC-system. Here is his summary:

"The two-part paper: Modular Hecke algebras and their Hopf symmetry; Rankin-Cohen brackets and the Hopf algebra of transverse geometry, (Moscow Math. J. 4 (2004), no. 1) exhibited several striking geometric features of the algebra generated by modular forms and Hecke operators. They all arose from the observation that the Hopf algebra encoding the transverse symmetries of codimension one foliations admits a natural action on this modular Hecke algebra and on its compressions.
The picture that emerged was that of a surprisingly close analogy with the transverse geometry of a codimension 1 foliation. This lecture outlines the construction of a Dirac twisted spectral triple, essentially obtained by replacing the usual Poincaré metric of the canonical bundle over the (connected component of the) Shimura variety with the Ramanujan metric defined in terms of the Dedekind eta function.
This type III spectral triple should provide the missing analytic framework for the underpinning geometry and explain its main attributes: the circle analogy, the arithmetic transgression of the Euler class, and the pseudodifferential calculus underlying the Rankin-Cohen deformations of the modular Hecke algebras."

Gunther Cornelissen's talk "Some Riemann Zeros for Function Fields" explained some computations for the space of toroidal automorphic forms. The following is his summary. "In the 1970s, Don Zagier suggested the following: a formula of Hecke proves that a GL_2 Eisenstein series of weight s integrates over a non-split torus to (almost) the zeta function of the corresponding quadratic field extension. Therefore, if we consider the space T of automorphic forms such that all these integrals vanish, and this space has a unitary structure, then the Riemann hypothesis (and simplicity of the zeros) would follow. I compute this space of so-called toroidal automorphic forms for the function field of three particular elliptic curves over finite fields. Here, the Riemann hypothesis is easily verified using the point-counting interpretation of the zeta function. In the talk, however, no point counting interpretation occured. Instead, only the Tate integral definition of zeta has been used. The condition that the toroidal integral vanishes for the constant extension torus is translated into an infinite system of linear equations in values of the automorphic form at the vertices of a Bruhat-Tits tree (for this, we need to iterate with Hecke operators). By solving this system and simultaneously the eigenvalue equation for the adjacency operator on the tree, we find that only 3 non-cusp eigenforms can occur as possible toroidal eigenforms. Since we know the eigenvalues of Eisenstein series, we can conclude that the Riemann hypothesis holds by comparing these with the possible toroidal eigenvalues.
If one does this computation for a rational function field, one can prove that its zeta function doesn't have any zeros. Now that is cracking a nut with a sledge hammer.
By some more advanced arguments (admissibility theorem of Winnie Li, integrating over other specific tori, multiplicity one for cusp forms, higher degree Hecke operators), one can even prove that in these three cases, the space of toroidal forms is one dimensional, spanned by the Eisenstein series of weight a zero of the corresponding zeta function.
References:

- paper is on arxiv:0710.2994

- thesis of Oliver Lorscheid, available from o.lorscheid@gmx.de (in this thesis, the space of toroidal forms is studied in general, and specifically computed for elliptic curves, but by a method that is a bit different, and in particular does not allow one to prove the Riemann hypothesis)."

Finally, in his talk: "Algebraic Continuation of multiple zeta values by Renormalization" Li Guo gave a remarkable application of the algebraic Birkhoff decomposition method of renormalization to the study of multiple zeta values outside the natural domain of convergence of the multiple series involved. His talk is available here.

NCG and F_un

(by A.C. and K. Consani)

Right after the end of the Sixth Annual Spring School/Conference on Noncommutative Geometry and Operator Algebras, a second meeting took place at Vanderbilt University, on May 15-18. This workshop has been dedicated to explore some aspects of several emerging relations linking Noncommutative Geometry and the geometry over the field with one element.
In the next weeks, we expect to post a more elaborate and thoughtful overview on this new interesting development in noncommutative/arithmetic geometry. In the meanwhile, the following is a first outline of the topics covered in several of the main talks. Some of the speakers have supplied us with the files of their transparencies and/or with an abstract of their presentation. When available, we added the abstracts within quotation marks "... ". The pdf files presently available have also been included here below.

The first day of the meeting was dedicated to the review and the discussion of the following 5 papers whose main subject has an evident connection with the field with one element.

1) Lisa Carbone gave in her talk "Kac-Moody groups, finite fields and Tits geometries" an overview of the seminal paper by Jacques Tits "Sur les analogues algebriques des groupes semi-simples complexes" (Colloque d'Algebre superieure, 1956, Bruxelles). The following is her review.
"Motivated by trying to find a "geometric" interpretation of a finite dimensional simple Lie group G in contrast to the "algebraic" version of G proposed by Chevalley the previous year, Tits introduced a "geometry" X which has G as its automorphism group. Tits' geometry X also had the mysterious property that when constructed over a finite field F_q , one could take the limit q --> 1 in which the group G tends to the discrete subgroup W (the Weyl group of G) and the geometry X tends to the geometry of W.
Tits' examples are quite sketchy in the 1956 paper. I would like to propose however that this was the seed of a deep circle of ideas that Tits cultivated and developed over three decades. The 1956 paper seems to have been a precursor to the notion of a Bruhat-Tits building for a Chevalley group over a finite field, or a simple algebraic group over a nonarchimedean local field. These constructions also evolve naturally into the notion of a Tits building for a Kac-Moody group over a finite field associated to the Tits functor for Kac-Moody groups.
In all of these subsequent constructions of Tits, the notion of a "field with 1 element" is present, both on the group level, and in the associated Tits geometry.
In my talk, I attempted to give an overview of examples from each of the classes described above, and to indicate what happens as we try to take a limit F_q --> F_1. This viewpoint has been very useful in my work, and I indicated a number of things I have been able to prove using the Tits geometries over F_1."

2) Christophe Soule review in his talk "Algebraic varieties over F_1" the main aspects developed in his paper "Les varietes sur le corps a un element" (Mosc. Math. J. 4 (2004), no. 1, 217--244, 312). An abstract of his presentation is downloadble here and in clear below:

Define a gadget over F_1 to be the pair X=(X, A_X) where X is a covariant functor from the category F of finite abelian groups to the category of sets, and A_X is a complex algebra. Given a finite abelian group G, a point x in X(G), and a character s of G, we assume given a character, e_{x,s} of A_X. If f: G-> G' is a morphism and y belongs to X (G'), the following equality is supposed to be satisfied : e_{f(y),s}= e_{y , s \circ f} for any character s.

An affine variety V over Z defines a gadget X over F_1 by letting X(G) be the set of points of V in the group algebra of G and by defining the algebra A_X to be the ring of regular functions on the complex points of V (with the obvious evaluation maps).

A morphism X -> Y, between two gadgets over F_1 consists of a natural transformation from the functor X to Y and a morphism of algebras from A_Y to A_X compatible with evaluation maps. It is called an immersion when both maps are injective.
An affine variety over F_1 is a gadget X such that
- For every G the set X(G) is finite;
- The complex algebra A_X is a commutative Banach algebra;
- There exists an affine variety X_Z over Z and an immersion i: X -> X_Z of gadgets satisfying the following property:
for any affine variety V over Z and any morphism of gadgets h: X -> V, there exists a unique algebraic morphism h_Z: X_Z -> V such that h equals h_Z composed with i.

Examples of varieties X_Z, where X is an affine variety over F_1, include smooth toric varieties and the algebraic group-schemes GL_2 and GL_3.

3) Niranjan Ramachandran gave in his talk "Zeta functions and motives (d'apres Manin)" an overview of the paper by Y. Manin "Lectures on zeta functions and motives (according to Deninger and Kurokawa)" (Columbia University Number-Theory Seminar, New-York 1992, Asterisque No. 228 (1995), 4, 121--163).
The following is his review.
"The aim of my talk was to provide a brief introduction to the beautiful paper of Yuri Manin (Lectures on motives and zeta functions - to be found on Katia's website www.math.jhu.edu/~kc) on the fascinating ideas of Christopher Deninger and Nobushige Kurokawa on zeta functions and F1.
The basic analogy between number fields and function fields has driven much of 20th century arithmetic geometry. This leads one to the desire to view Spec Z as a curve, but over which field? Of course, F1.
Deninger has expressed the completed Riemann zeta as R divided by s.(s-1)
where R is a regularized determinant to be viewed as infinite-dimensional analogue of a determinant of an endomorphism of a finite dimensional vector space. Compare with the zeta function of a smooth projective curve (of genus g) over a finite field F_q: a polynomial of degree 2g divided by (1-t) (1-qt) where t is the variable q^{-s}.
Manin provides an overview of the theory of motives over a finite field. He comments that even though we may not be able to define F1 or the category of varieties or motives over F1, we can certainly discuss zeta functions of motives over F1. The discussion strongly suggests that the only zeta functions that one obtains are generated by (s-n) for an integer n. Classical groups G are supposed to define varieties over F1 (original insight of Jacques Tits that G(F1) = W_G the Weyl group of G) as are projective spaces P^n; the zeta function of P^n over F1 is supposed to be s.(s-1)....(s-n). Thus the denominator in Deninger's expression for the completed Riemann zeta function is the zeta function of P1 over F1 which exactly parallels the function field case.
Manin also points out that the stable homotopy groups of spheres should be viewed as the algebraic K-theory of F_1 and the classical map J in algebraic topology from the stable homotopy groups to the algebraic K-theory of the integers is the one induced by the map F1 --> Z. This is a very important observation. The order of the image of J involves Bernoulli numbers (and hence zeta values!).
Manin also discusses the Kurokawa product of zeta functions and provides many examples from arithmetic and geometry (Selberg zeta, multiple gamma functions, ..) which could not be covered in this lecture. In particular, Kurokawa has defined the zeta function of (Spec Z) x_{Spec F1} (Spec Z) even though a mathematical definition of the fibre product is still lacking."

4) Jack Morava presented in his talk "K-theory of ring objects in homotopy theory" some relevant aspects of the paper by D. Quillen "On the cohomology and K-theory of the general linear groups over a finite field" (Annals of Mathematics, 2nd Ser., Vol.96, No.3, 1972, 552--586). An overview of his presentation is downladable here. Here is his abstract:

"Direct sum gives the category of finitely generated projective modules over a ring R (together with their isomorphisms) a symmetric monoidal structure. In 1972, Quillen defined the algebraic K-theory of R in terms of the best approximation to the geometric realization of this category by an abelian object in the homotopy category: an infinite loop-space or, in topologists' contemporary language, a spectrum.
These ideas have been vastly extended in the four decades since, in particular to general symmetric monoidal categories (Segal, eg finite sets) or to `categories with cofibrations and weak equivalences' (Waldhausen, eg finite cell complexes). Relatively recent developments (eg the theory of symmetric spectra) in our understanding of commutative ring objects in homotopy theory provide a unified approach to these generalizations and to related constructions (eg `topological' Hochschild and cyclic (co)homology).
My talk was basically historical; I tried to sketch the development of this language, and to use it to compare the category of vector spaces over a finite field and the category of finite sets. I wanted to clarify the extent to which the K-theory of the latter can be viewed as a limit, as q->1, of the K-theory of F_q."

5) Eugene Ha lectured on the main parts of the theory developed by Nikolai Durov in his preprint "New Approach to Arakelov Geometry" (arXiv:0704.2030). Here is the Eugene's review:

"Arakelov geometry is an amalgam of scheme-theoretic algebraic geometry and complex differential geometry that allows one to do intersection theory on models of algebraic varieties over the "compactification" of Spec(Z). However, missing from Arakelov's theory is a direct definition of the fiber at archimedean infinity, and in particular, of the notion of an "archimedean valuation ring." Recently, this situation has been rectified by Nikolai Durov who has created a full-fledged theory of generalized (commutative) rings and schemes that provides a framework for treating compactified arithmetic varieties in an uniform scheme-theoretic manner [A New Approach to ArakelovGeometry, 2007]. The "classical" algebraic geometry of Grothendieck is preserved in Durov's theory since the category of generalized rings contains the classical commutative rings as a full subcategory.
To develop a theory of "rings," like the "localization of Z at the infinity prime," that are monoidal but not additive, one can try to first frame classical ring theory in categorical terms,which has the advantage of allowing one to think of additivity as a monoidal structure. (For example, a ring is simply a monoid in the monoidal category of abelian groups.) It is well-known that the categorical notion that enables this transition is that of a monad in sets, i.e., a monoid in the monoidal category of endofunctors of sets. For a (classical) commutative ring R, the monad M_R attached to R is the functor that maps a set S to the set underlying the free R-module generated by S. The category of R-modules is then the category of modules of the monad M_R, and the ring R itself can be recovered from this category in the usual way (take the center of the endomorphism ring of the identity functor of R-modules).
This motivates the definition of a generalized commutative ring as a monad in sets which is moreover algebraic (commutes with filtered inductive limits of sets) and commutative (an algebraic monad in sets determines a family of n-ary operations on its modules, and commutativity for the monad means, roughly, that all these n-ary operations commute).
To see how one might arrive at a "correct" notion of the "local ring of Z at infinity" (or rather of its completion) suitable for a scheme-theoretic Arakelov geometry, Durov considers the notion of a "Z_infinity-lattice" in a real vector space. In the p-adic case, Z_p-lattices in a finite-dimensional p-adic vector space V correspond (up to similitude) to the maximal compact submonoids of End(V). This leads to the definition of Z_infinity-lattices (again, up to similitude) in a finite-dimensional real vector space E as the compact convex symmetric bodies in E. Further comparison with the p-adic case leads to the definition of the set underlying the free Z_infinity-module with basis S as the standard octahedron in R^{(S)}, and hence to the definition of Z_infinity as the generalized ring corresponding to this endofunctor.
In particular, Z_infinity is a generalized subring (i.e., algebraic submonad) of the real numbers R, as is Z_+, the generalized ring that maps a set S to the set of formal finite non-negative-integral linear combinations of elements of S. Thus one can take the intersection of Z_+ and Z_infinity: this is Durov's definition of F_1, the so-called field of one element. One can also describe F_1 as the free algebraic monad in sets with a single 0-"arity" generator. Modules over F_1 are simply sets with a marked point.
Going far beyond generalized commutative algebra, Durov has also developed a rather complete theory of spectra and generalized schemes. In his theory the "affine line" Spec(Z) is an affine scheme defined over F_1, and the compactification of Spec(Z) is a pro-generalized scheme.
Finally, while many of the motivations and constructionsof Durov's theory are very natural, the results of some of his computations differ from various widely-held expectations. Forexample, Durov has computed the Picard group of the compactification of Spec(Z) and has found that it is the multiplicative group of positive rational numbers, whereas the function field-number field analogy suggests that it should be be the positive real numbers. Moreover, the product S of Spec(Z) with itself over F_1 is shown to be Spec(Z) in Durov's theory, which is inconsistent with Kurokawa's definition of the zeta function of S."

Vanderbilt Lectures

The Sixth Annual Spring Institute on Noncommutative Geometry and Operator Algebras

in Vanderbilt University wrapped up yesterday. Some of the lectures and lecture series are already available online here.

On Gelfand-Naimark Theorems

There seems to be some inconsistency, mainly in the Internet, in naming one of the main theorems proved by Gelfand and Naimark in their foundational 1943 paper:

• On the imbedding of normed rings into the ring of operators in Hilbert space. Rec. Math. [Mat. Sbornik] N.S. 12(54), (1943). 197--213
  

.

I have seen the result in question referred to as Gelfand-Naimark' or Gelfand's theorem. Also, talking to younger people I get a sense of confusion as to how it should be called. Before getting to the theorem in question, let me indicate why this paper is so important. This paper is fundamental for the following 4 reasons:

1) C*-algebras were abstractly defined in this paper for the first time ever (in their main axioms they had two extra conditions, which as authors themselves indicated, but were unable to prove, were redundant. It took some 17 years to reach to the current concise formulation of the main axiom, called the C*-identity-see below. This needs another post to explain and I hope we can get to that in another time. This book gives a detailed account of this circle of ideas). Together with Murray-von Neumann's series of papers on Rings of Operators, or what later came to be called von Neumann algebras (1936-1943), the Gelfand-Naimark paper formed the foundaton stone of operator algebras and, eventually, noncommutative geometry.

2) Commutative C*-algebras were fully characterized in this paper as algebras of continuous functions on compact spaces. This is the theorem that concerns us in this post and we shall get to that later.


3) General (i.e. not necessarily commutative) C*-algebras were shown to admit a faithful embedding in the algebra of bounded operators on a Hilbert space.

4) The notion of state on a C*-algebra was introduced (but not under its current name) and used in the proof of 3) . This was later streamlined by I. Segal in 1947 and today we talk of the GNS (Gelfand-Naimark-Segal) construction.

Now the theorem in item 2) above (let us call it CGNT for `commutatve Gelfand-Naimark theorem') only appears as a Lemma in the paper, in page 3, and was not even mentioned in the introduction! For sure it was needed for the proof of the noncommutative theorem, but they also mention that it is of independent interst as well. Obviously the authors felt, as reflected in their title, that the noncommutative result in the main theorem of the paper. I have seen CGNT referred to as the Gelfand-Naimark theorem or Gelfand's isomorphism theorem. Operator algebra books correctly call it the commutative Gelfand-Naimark theorem. Wickepdia calls it Gelfand's reprsenation theorem and reseves Gelfand-Naimark for the noncommutative theorem in item 3) above.


The proof of the CGNT is based on Gelfand's theory of commutative Banach algebras and in fact is one of its landmark applications. Another, earlier, major success of the theory was Gelfand's surprizingly short and elegant proof of Wiener's 1/f theorem: if a function f has an absolutely convergent Fourier series and is nowhere zero then its inverse 1/f has an absolutely convergent Fourier series as well.

A (complex, unital) Banach algebra is a complex algebra, equipped with a complete normed vector space structure. Futhermore, the norm and the multipcative structure are related by the identity . A basic commutative example to keep in mind is the algebra C(X) of complex valued continuous functions on a compact Hausdorff space under pointwise addition and multiplication and the sup norm. But there are many other commutative examples of different natures. Important notions introduced in Gelfand's theory were that of the spectrum of an element and the fact that it is always non-empty; the spectrum of the algebra; and the Gelfand transform. The spectrum Spec (A) consists of multiplicative linear maps A....> C (charcaters or C-points of A). Being a subset of the dual space of A, it inherits a natural compact Hausdorff topology. Spec (A) can also be described as the set of maximal ideals of A: to a character associate its kernel...... The Gelfand transform is the more or less tautological map:

It is clearly an algebra map and is contractive (norm decreasing), but it need not be faithful. In the special case when A is the group algebra of an abelian group it reduces to Fourier transform. Finding the right class of commutative Banach algebras for which is an isometric isomorphism is what is acheived by CGNT. A C*-algebra is an involutive Banach algebra which satisfies the C*-identity

It is hard to exagerate the importance of the C*-identity. It has many implications, e.g. the uniqueness of the C*- norm, continuity of involutive algebra maps,....... Typically, for an involutive Banach algebra we have just an inequality x*x\leq x^2. The C*-identity puts
C*-algebras in a very special place among all Banach algebras, rather similar to the privileged position of Hilbert spaces among all Banach spaces. The world of Banach spaces is wild, but the Hilbertian universe is tame!

A few comments are in order here:

i) These days everything must be `categorical' (I am afraid this is utterly out of date now and I should say `categorified'!-but let us be pedantic). In fact the CGNT goes a long way towards establishing an equivalence between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Let us call these categories (with appropriate notion of morphism in each case) A and S. We have two functors
Spec: A^o .......>S and C: S ^o..............>A

(o means the dual or opposite category), assigning the spectrum and the algebra of complex valued continuous functions, respectively. These are equivalences of categories, (quasi) inverse of each other. In fact the composition C Spec: A .....> A is just the Gelfand transform and by the CGNT we know that it is isomorphic to the identity functor. This is the hard part. To show that the functor Spec C: S .......>S is isomorhic to the identity functor is much easier and is elementary. You just have to show that the Spec (C(X))=X for any X.

Now this way of thinking about the CGNT makes it very similar to other duality theorems in mathematics that puts in duality a category of spaces with a category of commutative algebras. A grand example of this is Hilbert's Nullstellensatz which implies that: the category of affine algebraic varieties over an algebraically closed field is equivalent to the dual of the category of finitely generated commutative reduced algebras. (reduced means there are no nilpotent elements). I think, comparing the two theorems, this reduced condition should be compared with the C*-identity. Here is a question that has puzzled me for some time and is for experts in noncommutative algebraic geometry: what is the right notion of a noncommutative affine algebraic variety sugested by Nullstellensatz? We know that in NCG the category of C*-algebras is in many ways a good category of noncommutative spaces. In other words we keep the C*-identity. In the algebraic case shall we keep this reduced condition?

ii) The spectrum of a commutative C* -algebra may not be `visible' at first sight, unless the algebra is already in the form C (X). This often leads to interesting results. For eaxmple, the algebra C_b (R) of bounded continuous functions on the real line R is a unital commutative
C*-algebra in a natural way. So, by CGNT , we know that C_b (R) = C ( X), where X is a compact Hausdorff space. What is X and how is it related to R? It is easy to see that is in fact the Stone-Cech compactification of R. More generally, for a locally compact Hausdorff space X the spectrum of C_b (X) can be shown to be homeomorphic to , the Stone-Cech compactification of X. For an example of a different flavour, let X be a topological space which is manifestly non-Hausdorff and let A=C (X). Then the spectrum of A has the effect of turning X into a Hausdorff space and is in some sense the `Hausdorffization' of X. The reader should try to describe the spectrum of .

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!

HEART BIT # 1

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