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?

John Conway once wrote :

There are almost as many different constructions of as there have been mathematicians interested in that most remarkable of all finite groups.

In the inguanodon post Ive added yet another construction of the Mathieu groups and starting from (half of) the Farey sequences and the associated cuboid tree diagram obtained by demanding that all edges are odd. In this way the Mathieu groups turned out to be part of a (conjecturally) infinite sequence of simple groups, starting as follows :

It is quite easy to show that none of the other sporadics will appear in this sequence via their known permutation representations. Still, several of the sporadic simple groups are generated by an element of order two and one of order three, so they are determined by a finite dimensional permutation representation of the modular group and hence are hiding in a special polygonal region of the Dedekind’s tessellation

Let us try to figure out where the sporadic with the next simplest permutation representation is hiding : the second Janko group , via its 100-dimensional permutation representation. The Atlas tells us that the order two and three generators act as

e:= (1,84)(2,20)(3,48)(4,56)(5,82)(6,67)(7,55)(8,41)(9,35)(10,40)(11,78)(12, 100)(13,49)(14,37)(15,94)(16,76)(17,19)(18,44)(21,34)(22,85)(23,92)(24, 57)(25,75)(26,28)(27,64)(29,90)(30,97)(31,38)(32,68)(33,69)(36,53)(39,61) (42,73)(43,91)(45,86)(46,81)(47,89)(50,93)(51,96)(52,72)(54,74)(58,99) (59,95)(60,63)(62,83)(65,70)(66,88)(71,87)(77,98)(79,80);

v:= (1,80,22)(2,9,11)(3,53,87)(4,23,78)(5,51,18)(6,37,24)(8,27,60)(10,62,47) (12,65,31)(13,64,19)(14,61,52)(15,98,25)(16,73,32)(17,39,33)(20,97,58) (21,96,67)(26,93,99)(28,57,35)(29,71,55)(30,69,45)(34,86,82)(38,59,94) (40,43,91)(42,68,44)(46,85,89)(48,76,90)(49,92,77)(50,66,88)(54,95,56) (63,74,72)(70,81,75)(79,100,83);

But as the kfarey.sage package written by Chris Kurth calculates the Farey symbol using the L-R generators, we use GAP to find those

L = e*v^-1  and  R=e*v^-2 so

L=(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)

R=(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)

Defining these permutations in sage and using kfarey, this gives us the Farey-symbol of the associated permutation representation

L=SymmetricGroup(Integer(100))("(1,84,22,46,70,12,79)(2,58,93,88,50,26,35)(3,90,55,7,71,53,36)(4,95,38,65,75,98,92)(5,86,69,39,14,6,96)(8,41,60,72,61,17, 64)(9,57,37,52,74,56,78)(10,91,40,47,85,80,83)(11,23,49,19,33,30,20)(13,77,15,59,54,63,27)(16,48,87,29,76,32,42)(18,68, 73,44,51,21,82)(24,28,99,97,45,34,67)(25,81,89,62,100,31,94)")

R=SymmetricGroup(Integer(100))("(1,84,80,100,65,81,85)(2,97,69,17,13,92,78)(3,76,73,68,16,90,71)(4,54,72,14,24,35,11)(5,34,96,18,42,32,44)(6,21,86,30,58, 26,57)(7,29,48,53,36,87,55)(8,41,27,19,39,52,63)(9,28,93,66,50,99,20)(10,43,40,62,79,22,89)(12,83,47,46,75,15,38)(23,77, 25,70,31,59,56)(33,45,82,51,67,37,61)(49,64,60,74,95,94,98)")

sage: FareySymbol("Perm",[L,R])

[[0, 1, 4, 3, 2, 5, 18, 13, 21, 71, 121, 413, 292, 463, 171, 50, 29, 8, 27, 46, 65, 19, 30, 11, 3, 10, 37, 64, 27, 17, 7, 4, 5], [1, 1, 3, 2, 1, 2, 7, 5, 8, 27, 46, 157, 111, 176, 65, 19, 11, 3, 10, 17, 24, 7, 11, 4, 1, 3, 11, 19, 8, 5, 2, 1, 1], [-3, 1, 4, 4, 2, 3, 6, -3, 7, 13, 14, 15, -3, -3, 15, 14, 11, 8, 8, 10, 12, 12, 10, 9, 5, 5, 9, 11, 13, 7, 6, 3, 2, 1]]

Here, the first string gives the numerators of the cusps, the second the denominators and the third gives the pairing information (where [tex[-2[/tex] denotes an even edge and an odd edge. Fortunately, kfarey also allows us to draw the special polygonal region determined by a Farey-symbol. So, here it is (without the pairing data) :

the hiding place of …

It would be nice to have (a) other Farey-symbols associated to the second Janko group, hopefully showing a pattern that one can extend into an infinite family as in the inguanodon series and (b) to determine Farey-symbols of more sporadic groups.

MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk on MUBs (as well as an old talk by Wootters).

So what are MUBs to mathematicians? Recall that a d-state quantum system is just the vectorspace equipped with the usual Hermitian inproduct . An observable is a choice of orthonormal basis consisting of eigenvectors of the self-adjoint matrix . together with another observable (with orthonormal basis ) are said to be mutally unbiased if the norms of all inproducts are equal to . This definition extends to a collection of pairwise mutually unbiased observables. In a d-state quantum system there can be at most d+1 mutually unbiased bases and such a collection of observables is then called a MUB of the system. Using properties of finite fields one has shown that MUBs exists whenever d is a prime-power. On the other hand, existence of a MUB for d=6 still seems to be open…

The King’s Problem¹ is the following : A physicist is trapped on an island ruled by a mean king who promises to set her free if she can give him the answer to the following puzzle. The physicist is asked to prepare a d−state quantum system in any state of her choosing and give it to the king, who measures one of several mutually unbiased observables on it. Following this, the physicist is allowed to make a control measurement on the system, as well as any other systems it may have been coupled to in the preparation phase. The king then reveals which observable he measured and the physicist is required to predict correctly all the eigenvalues he found.

The Solution to the King’s problem in prime power dimension by P. K. Aravind, say for , consists in taking a system of k object qupits (when one qupit is a spin l particle) which she will give to the King together with k ancilla qupits that she retains in her possession. These 2k qupits are diligently entangled and prepared is a well chosen state. The final step in finding a suitable state is the solution to a pure combinatorial problem :

She must use the numbers 1 to d to form ordered sets of d+1 numbers each, with repetitions of numbers within a set allowed, such that any two sets have exactly one identical number in the same place in both. Here’s an example of 16 such strings for d=4 :

11432, 12341, 13214, 14123, 21324, 22413, 23142, 24231, 31243, 32134, 33421, 34312, 41111, 42222, 43333, 44444

Here again, finite fields are used in the solution. When , identify the elements of with the numbers from 1 to d in some fixed way. Then, the of number-strings are found as follows : let and take as the first 2 numbers the ones corresponding to these field-elements. The remaning d-2 numbers in the string are those corresponding to the field element (with ) determined from by the equation

where is the field-element corresponding to the integer i ( corresponds to the zero element). It is easy to see that these strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine (and hence the whole string) as it follows from and that .

In the special case when d=3 (that is, one spin 1 particle is given to the King), we recover the tetracode : the nine codewords

0000, 0+++, 0—, +0+-, ++-0, +-0+, -0-+, -+0-, –+0

encode the strings (with +=1,-=2,0=3)

3333, 3111, 3222, 1312, 1123, 1231, 2321, 2132, 2213

1. actually a misnomer, it’s more the poor physicists’ problem… [↩]

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

is a noncommutative algebra, the Bost-Connes Hecke algebra , which is a bi-chrystalline graded algebra (somewhat weaker than ’strongly graded’) with part of degree one the group-algebra . Further, has a natural one-parameter family of algebra automorphisms defined by and .

For any algebra together with a one-parameter family of automorphisms one is interested in KMS-states or Kubo-Martin-Schwinger states with parameter , (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 on (in particular it must have norm one) and it belongs to if the following commutation relation holds for all elements

Let us work out the special case when is the matrix-algebra . To begin, all algebra-automorphisms are inner in this case, so any one-parameter family of automorphisms is of the form

where is the matrix-exponential of the nxn matrix . For any parameter we claim that the linear functional

is a KMS-state.Indeed, we have for all matrices that

(the next to last equality follows from cyclic-invariance of the trace map). These states are usually called Gibbs states and the normalization factor (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 : take a simple n-dimensional representation , find the matrix determining the image of the automorphisms and take the Gibbs states as defined before.

Let us return now to the Bost-Connes algebra . We don’t know any finite dimensional simple representations of 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 (all of them being one-dimensional and corresponding to characters of ).

Hence, for any ( details) we have a graded simple -representation with action defined by

Here, is computed using the ‘chinese-remainder-identification’ ( details).

Even when the representations are not finite dimensional, we can mimic the above strategy : we should find a linear operator determining the images of the automorphisms . We claim that the operator is defined by for all . That is, we claim that for elements we have

So let us compute the action of both sides on when . The left hand side gives whereas the right-hand side becomes

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

Finally, let us calculate the normalization factor (or partition function) . Because we have for that the trace

is equal to the Riemann zeta-value (at least when ).

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

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: 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 are given by and .

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.