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New world record obscurification

Monday, April 7th, 2008

I’ve always thought of Alain Connes as the unchallengeable world-champion opaque mathematical writing, but then again, I was proven wrong.

Alain’s writings are crystal clear compared to the monstrosity the AMS released to the world : In search of the Riemann zeros - Strings, fractal membranes and noncommutative spacetimes by Michel L. Lapidus.

Here’s a generic half-page from a total of 558 pages (or rather 314, as the remainder consists of appendices, bibliography and indices…). I couldn’t find a single precise, well-defined and proven statement in the entire book.

4.2. Fractal Membranes and the Second Quantization of Fractal Strings
“The first quantization is a mystery while the second quantization is a functor” Edward Nelson (quoted in [Con6,p.515])

We briefly discuss here joint work in preparation with Ryszard Nest [LapNe1]. This work was referred to several times in Chapter 3, and, as we pointed out there, it provides mathematically rigorous construction of fractal membranes (as well as of self-similar membranes), in the spirit of noncommutative geometry and quantum field theory (as well as of string theory). It also enables us to show that the expected properties of fractal (or self-similar) membranes, derived in our semi-heuristic model presented in Sections 3.2 and 3.2. are actually satisfied by the rigorous model in [LapNe1]. In particular, there is a surprisingly good agreement between the author’s original intuition on fractal (or self-similar) membrane, conceived as an (adelic) Riemann surface with infinite genus or as an (adelic) infinite dimensional torus, and properties of the noncommutative geometric model in [LapNe1]. In future joint work, we hope to go beyond [LapNe1] and to give even more (noncommutative) geometric content to this analogy, possibly along the lines suggested in the next section (4.3).
We will merely outline some aspects of the construction, without supplying any technical details, instead referring the interested reader to the forthcoming paper [LapNe1] for a complete exposition of the construction and precise statements of results.

Can the AMS please explain to the interested person buying this book why (s)he will have to await a (possible) forthcoming paper to (hopefully) make some sense of this apparent nonsense?

Tags: Connes, geometry, noncommutative, Riemann
Posted in geometry | 2 Comments »

KMS, Gibbs & zeta function

Thursday, February 21st, 2008

Noncommutative geometry and the Riemann zeta function

Time to wrap up this series on the Bost-Connes algebra. Here’s what we have learned so far : the convolution product on double cosets

$\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix} \backslash \begin{bmatrix} 1 & \mathbb{Q} \\ 0 & \mathbb{Q}_{> 0} \end{bmatrix} / \begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix}$

is a noncommutative algebra, the Bost-Connes Hecke algebra $\mathcal{H}$ , which is a bi-chrystalline graded algebra (somewhat weaker than ’strongly graded’) with part of degree one the group-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ . Further, $\mathcal{H}$ has a natural one-parameter family of algebra automorphisms $\sigma_t$ defined by $\sigma_t(X_n) = n^{it}X_n$ and $\sigma_t(Y_{\lambda})=Y_{\lambda}$ .

For any algebra together with a one-parameter family of automorphisms $\sigma_t$ one is interested in KMS-states or Kubo-Martin-Schwinger states with parameter $\beta$ , $KMS_{\beta}$ (this parameter is often called the ‘invers temperature’ of the system) as these are suitable equilibria states. Recall that a state is a special linear functional $\phi$ on (in particular it must have norm one) and it belongs to $KMS_{\beta}$ if the following commutation relation holds for all elements $a,b \in A$

$\phi(a \sigma_{i\beta}(b)) = \phi(b a)$

Let us work out the special case when is the matrix-algebra $M_n(\mathbb{C})$ . To begin, all algebra-automorphisms are inner in this case, so any one-parameter family of automorphisms is of the form

$\sigma_t(a) = e^{itH} a e^{-itH}$

where $e^{itH}$ is the matrix-exponential of the nxn matrix . For any parameter $\beta$ we claim that the linear functional

$\phi(a) = \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H})$

is a KMS-state.Indeed, we have for all matrices $a,b \in M_n(\mathbb{C})$ that

$\phi(a \sigma_{i \beta}(b)) = \frac{1}{tr(e^{-\beta H})} tr(a e^{- \beta H} b e^{\beta H} e^{- \beta H})$

$= \frac{1}{tr(e^{-\beta H})} tr(a e^{-\beta H} b) = \frac{1}{tr(e^{-\betaH})} tr(ba e^{-\beta H}) = \phi(ba)$

(the next to last equality follows from cyclic-invariance of the trace map). These states are usually called Gibbs states and the normalization factor $\frac{1}{tr(e^{-\beta H})}$ (needed because a state must have norm one) is called the partition function of the system. Gibbs states have arisen from the study of ideal gases and the place to read up on all of this are the first two chapters of the second volume of “Operator algebras and quantum statistical mechanics” by Ola Bratelli and Derek Robinson.

This gives us a method to construct KMS-states for an arbitrary algebra with one-parameter automorphisms $\sigma_t$ : take a simple n-dimensional representation $\pi~:~A \mapsto M_n(\mathbb{C})$ , find the matrix determining the image of the automorphisms $\pi(\sigma_t)$ and take the Gibbs states as defined before.

Let us return now to the Bost-Connes algebra $\mathcal{H}$ . We don’t know any finite dimensional simple representations of $\mathcal{H}$ but, sure enough, have plenty of graded simple representations. By the usual strongly-graded-yoga they should correspond to simple finite dimensional representations of the part of degree one $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ (all of them being one-dimensional and corresponding to characters of $\mathbb{Q}/\mathbb{Z}$ ).

Hence, for any $u \in \mathcal{G} = \prod_p \hat{\mathbb{Z}}_p^{\ast}$ ( details) we have a graded simple $\mathcal{H}$ -representation $S_u = \oplus_{n \in \mathbb{N}_+} \mathbb{C} e_n$ with action defined by

$\begin{cases} \pi_u(X_n)(e_m) = e_{nm} \\ \pi_u(Y_{\lambda})(e_m) = e^{2\pi i n u . \lambda} e_m \end{cases}$

Here, $u.\lambda$ is computed using the ‘chinese-remainder-identification’ $\mathcal{A}/\mathcal{R} = \mathbb{Q}/\mathbb{Z}$ ( details).

Even when the representations S_u are not finite dimensional, we can mimic the above strategy : we should find a linear operator determining the images of the automorphisms $\pi_u(\sigma_t)$ . We claim that the operator is defined by H(e_n) = log(n) e_n for all $n \in \mathbb{N}_+$ . That is, we claim that for elements $a \in \mahcal{H}$ we have

$\pi_u(\sigma_t(a)) = e^{itH} \pi_u(a) e^{-itH}$

So let us compute the action of both sides on e_m when a=X_n . The left hand side gives $\pi_u(n^{it}X_n)(e_m) = n^{it} e_{mn}$ whereas the right-hand side becomes

$e^{itH}\pi_u(X_n) e^{-itH}(e_m) = e^{itH} \pi_u(X_n) m^{-it} e_m =$

$e^{itH} m^{-it} e_{mn} = (mn)^{it} m^{-it} e_{mn} = n^{it} e_{mn}$

proving the claim. For any parameter $\beta$ this then gives us a KMS-state for the Bost-Connes algebra by

$\phi_u(a) = \frac{1}{Tr(e^{-\beta H})} Tr(\pi_u(a) e^{-\beta H})$

Finally, let us calculate the normalization factor (or partition function) $\frac{1}{Tr(e^{-\beta H})}$ . Because $e^{-\beta H}(e_n) = n^{-\beta} e_n$ we have for that the trace

$Tr(e^{-\beta H}) = \sum_{n \in \mathbb{N}_+} \frac{1}{n^{\beta}} = \zeta(\beta)$

is equal to the Riemann zeta-value $\zeta(\beta)$ (at least when $\beta > 1$ ).

Summarizing, we started with the definition of the Bost-Connes algebra $\mathcal{H}$ , found a canonical one-parameter subgroup of algebra automorphisms $\sigma_t$ and computed that the natural equilibria states with respect to this ‘time evolution’ have as their partition function the Riemann zeta-function. Voila!

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Tags: Artin, Connes, Gibbs, noncommutative, partition function, representations, Riemann, states, zeta function
Posted in geometry | 2 Comments »

“God given time”

Wednesday, February 20th, 2008

Noncommutative geometry and the Riemann zeta function

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

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

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure theory, which in essence only cares about the “quantity of points” in a space, one ﬁnds unexpected completely new features in the noncommutative case. While it had been long known by operator algebraists that the theory of von-Neumann algebras represents a far reaching extension of measure theory, the main surprise which occurred at the beginning of the seventies is that such an algebra M inherits from its noncommutativity a god-given time evolution: $\delta~:~\mathbb{R} \rightarrow Out(M)$ where is the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms. This led in my thesis to the reduction from type III to type II and their automorphisms and eventually to the classiﬁcation of injective factors.

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

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

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

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

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Tags: Artin, Connes, Cuntz, geometry, noncommutative, Quillen
Posted in geometry | 2 Comments »

BC stands for Bi-Crystalline graded

Saturday, January 26th, 2008

Noncommutative geometry and the Riemann zeta function

Towards the end of the Bost-Connes for ringtheorists post I freaked-out because I realized that the commutation morphisms with the X_n^* were given by non-unital algebra maps. I failed to notice the obvious, that algebras such as $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ have plenty of idempotents and that this mysterious ‘non-unital’ morphism was nothing else but multiplication with an idempotent…

Here a sketch of a ringtheoretic framework in which the Bost-Connes Hecke algebra $\mathcal{H}$ is a motivating example (the details should be worked out by an eager 20-something). Start with a suitable semi-group , by which I mean that one must be able to invert the elements of and obtain a group of which all elements have a canonical form $g=s_1s_2^{-1}$ . Probably semi-groupies have a name for these things, so if you know please drop a comment.

The next ingredient is a suitable ring . Here, suitable means that we have a semi-group morphism $\phi~:~S \rightarrow End(R)$ where End(R) is the semi-group of all ring-endomorphisms of satisfying the following two (usually strong) conditions :

Every $\phi(s)$ has a right-inverse, meaning that there is an ring-endomorphism $\psi(s)$ such that $\phi(s) \circ \psi(s) = id_R$ (this implies that all $\phi(s)$ are in fact epi-morphisms (surjective)), and
The composition $\psi(s) \circ \phi(s)$ usually is NOT the identity morphism (because it is zero on the kernel of the epimorphism $\phi(s)$ ) but we require that there is an idempotent $E_s \in R$ (that is, ) such that $\psi(s) \circ \phi(s) = id_R E_s$

The point of the first condition is that the -semi-group graded ring $A = \oplus_{s \in S} X_s R$ is crystalline graded (crystalline group graded rings were introduced by Fred Van Oystaeyen and Erna Nauwelaarts) meaning that for every $s \in S$ we have in the ring the equality X_s R = R X_s where this is a free right -module of rank one. One verifies that this is equivalent to the existence of an epimorphism $\phi(s)$ such that for all $r \in R$ we have $r X_s = X_s \phi(s)(r)$ .

The point of the second condition is that this semi-graded ring can be naturally embedded in a -graded ring $B = \oplus_{g=s_1s_2^{-1} \in G} X_{s_1} R X_{s_2}^*$ which is bi-crystalline graded meaning that for all $r \in R$ we have that $r X_s^* = X_s^* \psi(s)(r) E_s$ .

It is clear from the construction that under the given conditions (and probably some minor extra ones making everything stand) the group graded ring is determined fully by the semi-group graded ring .

what does this general ringtheoretic mumbo-jumbo have to do with the BC- (or Bost-Connes) algebra $\mathcal{H}$ ?

In this particular case, the semi-group is the multiplicative semi-group of positive integers $\mathbb{N}^+_{\times}$ and the corresponding group is the multiplicative group $\mathbb{Q}^+_{\times}$ of all positive rational numbers.

The ring is the rational group-ring $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ of the torsion-group $\mathbb{Q}/\mathbb{Z}$ . Recall that the elements of $\mathbb{Q}/\mathbb{Z}$ are the rational numbers $0 \leq \lambda < 1$ and the group-law is ordinary addition and forgetting the integral part (so merely focussing on the ‘after the comma’ part). The group-ring is then

$\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] = \oplus_{0 \leq \lambda < 1} \mathbb{Q} Y_{\lambda}$ with multiplication linearly induced by the multiplication on the base-elements $Y_{\lambda}.Y_{\mu} = Y_{\lambda+\mu}$ .

The epimorphism determined by the semi-group map $\phi~:~\mathbb{N}^+_{\times} \rightarrow End(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}])$ are given by the algebra maps defined by linearly extending the map on the base elements $\phi(n)(Y_{\lambda}) = Y_{n \lambda}$ (observe that this is indeed an epimorphism as every base element $Y_{\lambda} = \phi(n)(Y_{\frac{\lambda}{n}})$ .

The right-inverses $\psi(n)$ are the ring morphisms defined by linearly extending the map on the base elements $\psi(n)(Y_{\lambda}) = \frac{1}{n}(Y_{\frac{\lambda}{n}} + Y_{\frac{\lambda+1}{n}} + \hdots + Y_{\frac{\lambda+n-1}{n}})$ (check that these are indeed ring maps, that is that $\psi(n)(Y_{\lambda}).\psi(n)(Y_{\mu}) = \psi(n)(Y_{\lambda+\mu})$ .

These are indeed right-inverses satisfying the idempotent condition for clearly $\phi(n) \circ \psi(n) (Y_{\lambda}) = \frac{1}{n}(Y_{\lambda}+\hdots+Y_{\lambda})=Y_{\lambda}$ and

$\begin{eqnarray_} \psi(n) \circ \phi(n) (Y_{\lambda}) =& \psi(n)(Y_{n \lambda}) = \frac{1}{n}(Y_{\lambda} + Y_{\lambda+\frac{1}{n}} + \hdots + Y_{\lambda+\frac{n-1}{n}}) \\ =& Y_{\lambda}.(\frac{1}{n}(Y_0 + Y_{\frac{1}{n}} + \hdots + Y_{\frac{n-1}{n}})) = Y_{\lambda} E_n \end{eqnarray_}$

and one verifies that $E_n = \frac{1}{n}(Y_0 + Y_{\frac{1}{n}} + \hdots + Y_{\frac{n-1}{n}})$ is indeed an idempotent in $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ . In the previous posts in this series we have already seen that with these definitions we have indeed that the BC-algebra is the bi-crystalline graded ring

$B = \mathcal{H} = \oplus_{\frac{m}{n} \in \mathbb{Q}^+_{\times}} X_m \mathbb{Q}[\mathbb{Q}/\mathbb{Z}] X_n^*$

and hence is naturally constructed from the skew semi-group graded algebra $A = \oplus_{m \in \mathbb{N}^+_{\times}} X_m \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ .

This (probably) explains why the BC-algebra $\mathcal{H}$ is itself usually called and denoted in C^* -algebra papers the skew semigroup-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \bowtie \mathbb{N}^+_{\times}$ as this subalgebra (our crystalline semi-group graded algebra ) determines the Hecke algebra completely.

Finally, the bi-crystalline idempotents-condition works well in the settings of von Neumann regular algebras (such as all limits of finite dimensional semi-simples, for example $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ ) because such algebras excel at idempotents galore…

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Tags: Connes, crystalline, graded, noncommutative, semi-group, simples
Posted in geometry | 4 Comments »

yahoo pipes on iTouch

Thursday, January 24th, 2008

iTouch

The next thing on my tech-to-do-list : learn all about Yahoo Pipes :

Pipes is a powerful composition tool to aggregate, manipulate, and mashup content from around the web. Like Unix pipes, simple commands can be combined together to create output that meets your needs. Here are a few popular ways the service can be used:
- create your ultimate custom feed by combining many feeds into one, then sorting, filtering and translating them.
- geocode your favorite feeds and browse the items on an interactive map.
- remix your favorite data sources and use the Pipe to power a new application.
- build custom vertical search pages that are impossible with ordinary search engines.
- power widgets/badges on your web site.
- consume the output of any Pipe in RSS, JSON, KML, and other formats.

I’ve posted before on setting up your own lifestream, or your own planet, or scraping feeds, or subscribing to my brain, or … whatever. The good news is : all these ideas are now superseded by Pipes!

Pipes is a free online service that lets you remix popular feed types and create data mashups using a visual editor. You can use Pipes to run your own web projects, or publish and share your own web services without ever having to write a line of code. You make a Pipe by dragging pre-configured modules onto a canvas and wiring them together in the Pipes Editor. Once you’ve built a Pipe, you’ll be able save it on our server and then call it like you would any other feed. Pipes offers output in RSS 2.0, RSS 1.0 (RDF), JSON and Atom formats for maximum flexibility. You can also choose to publish your Pipe and share it with the world, allowing other users to clone it, add their own improvements, or use it as a subcomponent in their own creations.

This is the essential message to get : yahoo-pipes allows you to remix the web, filtering out all noise! And the good news is

There are plenty of public pipes around to get you going, and
Pipes has an iTouch-friendly interface (see above left). All you have to do is to Safari to iphone.pipes.yahoo.com and use them.

Here are a few public-pipes you can use out of the box!

iPhone / iPod Touch: The Most Comprehensive Feed Ever!, doing what it promises : giving you the best iTouch-posts without having to roam for them.
JSON Geocoder, returning lat/lon/address info from the the given address.
Uber Blog Search, Search all the blogosphere with one query. Hits Google, Ask, Technorati, and icerocket then returns the unique results. Below the web-interface giving the results for ‘noncommutative’…

Meta Search Alerts if you want a less ‘neverendingbooks’-based reply

and finally, one of my favorites, implementing to some extend the Lifestream-idea (iTouch-interface above left)

lifefeed - virable, Easily Aggregate your social whereabouts great for blogs profiles and more! Aggregates Your Feeds From: -Digg -Last.fm -Twitter -Flickr -Del.icio.us and your very own blog Adopt and Improve, enjoy!

I’ll promise to spend some time soon to set up my very own pipes and make them available…

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Tags: brain, GMD, google, itouch, noncommutative, pipes, sage, social, tech, web 2.0
Posted in iMath, mac | 5 Comments »

Posts Tagged ‘noncommutative’

New world record obscurification

KMS, Gibbs & zeta function

Noncommutative geometry and the Riemann zeta function

“God given time”

Noncommutative geometry and the Riemann zeta function

BC stands for Bi-Crystalline graded

Noncommutative geometry and the Riemann zeta function

yahoo pipes on iTouch

iTouch

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