I expressed several times the belief that sub-riemannian geometry represents an example of a mathematically new phenomenon, which I call “non-commutative analysis”. Repeatedly happened that apparently general results simply don’t work well when applied to sub-riemannian geometry. This “strange” (not for me) phenomenon leads to negative statements, like rigidity results (Mostow, Margulis), non-rectifiability results (like for example the failure of the theory of metric currents for Carnot groups). And now, to this adds the following, arXiv:1404.5494 [math.OA]
“the unexpected result that the theory of spectral triples does not apply to the Carnot manifolds in the way one would expect. [p. 11] ”
“We will prove in this thesis that any horizontal Dirac operator on an arbitrary Carnot manifold cannot be hypoelliptic. This is a big difference to the classical case, where any Dirac operator is elliptic. [p. 12]”
It appears that the author reduces the problems to the Heisenberg groups. There is a solution, then, to use
R. Beals, P.C. Greiner, Calculus on Heisenberg manifolds, Princeton University Press, 1988
which gives something resembling spectral triples, but not quite all works, still:
“and show how hypoelliptic Heisenberg pseudodifferential operators furnishing a spectral triple and detecting in addition the Hausdorff dimension of the Heisenberg manifold can be constructed. We will suggest a few concrete operators, but it remains unclear whether one can detect or at least estimate the Carnot-Caratheodory metric from them. [p. 12]”
This seems to be an excellent article, more than that, because it is a phd dissertation many things are written clearly.
I am not surprised at all by this, it just means that, as in the case with the metric currents, there is an ingredient in the spectral triples theory which introduces by the backdoor some commutativity, which messes then with the non-commutative analysis (or calculus).
Instead I am even more convinced than ever that the minimal (!) description of sub-riemannian manifolds, as models of a non-commutative analysis, is given by dilation structures, explained most recently in arXiv:1206.3093 [math.MG].
A corollary of this is: sub-riemannian geometry (i.e. non-commutative analysis of dilation structures) is more non-commutative than non-commutative geometry .
I’m waiting for a negative result concerning the application of quantum groups to sub-riemannian geometry.
A complete, locally compact riemannian manifold is a length metric space by the Hopf-Rinow theorem. The problem of intrinsic characterization of riemannian spaces asks for the recovery of the manifold structure and of the riemannian metric from the distance function coming from to the length functional.
For 2-dim riemannian manifolds the problem has been solved by A. Wald in 1935. In 1948 A.D. Alexandrov introduces his famous curvature (which uses comparison triangles) and proves that, under mild smoothness conditions on this curvature, one is capable to recover the differential structure and the metric of the 2-dim riemannian manifold. In 1982 Alexandrov proposes as a conjecture that a characterization of a riemannian manifold (of any dimension) is possible in terms of metric (sectional) curvatures (of the type introduced by Alexandrov) and weak smoothness assumptions formulated in metric way (as for example Hölder smoothness).
The problem has been solved by Nikolaev in 1998, in the paper A metric characterization of Riemannian spaces. Siberian Adv. Math. 9, no. 4 (1999), 1-58. The solution of Nikolaev can be summarized like this: he starts with a locally compact length metric space (and some technical details), then
- he constructs a (family of) intrinsically defined tangent bundle(s) of the metric space, by using a generalization of the cosine formula for estimating a kind of a distance between two curves emanating from different points. This will lead him to a generalization of the tangent bundle of a riemannian manifold endowed with the canonical Sasaki metric.
- He defines a notion of sectional curvature at a point of the metric space, as a limit of a function of nondegenerated geodesic triangles, limit taken as these triangles converge (in a precised sense) to the point.
- The sectional curvature function thus constructed is supposed to satisfy a Hölder continuity condition (thus a regularity formulated in metric terms)
- He proves then that the metric space is isometric with (the metric space associated to) a riemannian manifold of precise (weak) regularity (the regularity is related to the regularity of the sectional curvature function).
Sub-riemannian spaces are length metric spaces as well. Any riemannian space is a sub-riemannian one. It is not clear at first sight why the characterization of riemannian spaces does not extend to sub-riemannian ones. In fact, there are two problematic steps for such a program for extending Nikolaev result to sub-riemannian spaces:
- the cosine formula, as well as the Sasaki metric on the tangent bundle don’t have a correspondent in sub-riemannian geometry (because there is, basically, no statement canonically corresponding to Pythagoras theorem);
- the sectional curvature at a point cannot be introduced by means of comparison triangles, because sub-riemanian spaces do not behave well with respect to this comparison of triangle idea, as proved by Scott Pauls.
In 1996 M. Gromov formulates the problem of intrinsic characterization of sub-riemannian spaces. He takes the Carnot-Caratheodory (or CC) distance (this is the name of the distance constructed on a sub-riemannian manifold from the differential geometric data we have, which generalizes the construction of the riemannian distance from the riemannian metric) as the only intrinsic object of a sub-riemannian space. Indeed, in the linked article, section 0.2.B. he writes:
If we live inside a Carnot-Caratheodory metric space V we may know nothing whatsoever about the (external) infinitesimal structures (i.e. the smooth structure on , the subbundle and the metric on ) which were involved in the construction of the CC metric.
Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces.
Give a minimal, axiomatic, description of sub-riemannian spaces.
Continuing from Dictionary from emergent algebra to graphic lambda calculus (II) , let’s introduce the following macro, which is called “relative gate”:
If this macro looks involved, then we might express it with the help of emergent algebra crossing macros, like this:
Notice that none of these graphs are in the emergent algebra sector, a priori! However, by looking at the graph from the left, we recognize an old friend: a chora. See arXiv:1103.6007 section 5, for a definition of the chora construction, but only in relation with emergent algebra gates. Previously the chora diagram appeared as a convenient notation for the relative dilation gate, but here we see it appearing in a different context. It is already present in relation with lambda calculus in the lambda-scale article arXiv:1205.0139 , section 4. It will be important later, when I shall give an exposition of the second paragraph after Question 1 from the post Graphic lambda calculus used for quantum programming (Towards qubits III) .
With the help of the relative macro, we can give a new look at the mystery move from the end of dictionary II post. Indeed, look at the following succession of moves:
The use of the mystery move is to transform the relative gate into a usual gate! So, if we accept the mystery move for the emergent algebra sector, then the effect is that the relative gate is in the emergent algebra sector of the graphic lambda calculus and, moreover, it can be transformed into a gate.
With this information let’s go back to the graphs and from the dictionary II post.
We know that in the emergent algebra sector is the right translation of the approximate associativity from emergent algebras (formalism using binary trees, for example). In the last post we arrived at the conclusion that we can prove by using the mystery move as a legal move in the emergent algebra sector of the graphic lambda calculus (in combination with the other valid moves for this sector, but not using GOBAL FAN-OUT).
Instead of the graph , we may introduce the graph and compare it with the graph :
The only difference between and is given by the appearance of the relative gate, in , instead of the regular gate in .
We can prove that . In particular this proves that is in the emergent algebra sector, even if it contains a relative gate. Recall that if we don’t accept the mystery move in the emergent algebra sector, then it’s not clear if it belongs to that sector. The proof is not given, but it’s straightforward, using the moves CO-ASSOC, CO-COMM, LOC PRUNING, R2 and ext 2 (therefore without the mystery move).
We can pass from to by using the fact that the mystery move allows to transform a relative gate into a regular one. So this is the way in which enters the mystery move in the equivalence : through , which can be done without the mystery move, and by the mystery move, under the form of transforming a relative gate into a regular one.
This post continues the Dictionary from emergent algebra to graphic lambda calculus (I). Let us see if we can prove the approximate associativity property (for the approximate sum) in graphic lambda calculus. I am going to use the dictionary and see if I can make the translation.
With the dilation structures/tree formalism, the approximate associativity, with it’s proof, is this:
Let’s translate the trees, by using the dictionary.
At first sight, the graphs and are clearly in the emergent algebra sector of the graphic lambda calculus. For the graph , which corresponds to the tree from the middle of the first figure, it’s not clear if it belongs to the same sector.
The next figure shows that it does, because we can pass from to by a succession of moves acceptable in the sector.
Problems appear when we try to relate and . The reason of these problems, we shall see, is the fact that we renounced at having variables, by employing the “fan-out” gate . But this gate is a fan-out only in a very weak sense, in fact is more like a co-commutative co-monoid operation.
Let’s see, we prepare in order to be more alike .
There is a part of the final graph which is encircled by a green dashed line, pay attention to it.
We prepare now a bit:
The graph from the right has also an encircled part in green. Look close to this graph, in parallel with the last graph from the preparation of . We remark that these two graphs are the same outside of the respective encircled regions. If we could transform one encircled region into the other then we would have a proof that .
In conclusion, the mystery move
would solve the problem of proving the approximate associativity in the emergent algebra sector of the graphic lambda calculus. Remark that if would really be a fan-out gate, i.e. if we could apply to it GLOBAL FAN-OUT moves, then the mystery move would be true.
But I don’t want to use global moves in the emergent algebra sector, because in the tree formalism all moves are local. Moreover, even with global fan-out, in order to be able to apply it, it would be necessary that at the labels “x” and “u” to be grafted graphs which have no connection in common, therefore, strictly speaking, we succeed to prove an approximate associativity weaker than the approximate associativity from emergent algebras/tree formalism. (The same phenomenon, but for another identity of emergent algebras, is explained in this post.) The reason is in the apparently innocuous but hard to manage fact that we renounce of having variables in graphic lambda calculus.
Next time, about the meaning of the mystery move.
Because I am going to explore in future posts the emergent algebra sector, I think it is good to know where we stand with using graphic lambda calculus for describing proofs in emergent algebra as computations. In the big map of research paths, this correspond to the black path linking “Energent algebra sector” with “Emergent algebras”.
A dictionary seems a good way to start this discussion.
Let’s see, there are three formalisms there:
- in the first paper on spaces with dilations, Dilatation structures I. Fundamentals arXiv:math/0608536 section 4, is introduced a formalism using binary decorated trees in order to ease the manipulations of dilation structures,
- emergent algebras are an abstraction of dilation structures, in the sense that they don’t need a metric space to live on. The first paper on the subject is Emergent algebras arXiv:0907.1520 (see also Braided spaces with dilations and sub-riemannian symmetric spaces arXiv:1005.5031 for explanations of the connection between dilation structures and emergent algebras, as well as for braided symmetric spaces, sub-riemannian symmetric spaces, conical groups, items you can see on the big map mentioned before). Emergent algebras is a mixture of an algebraic theory with an important part of epsilon-delta analysis. One of the goals of graphic lambda calculus is to replace this epsilon-delta part by a computational part.
- graphic lambda calculus, extensively described here, has an emergent algebra sector (see arXiv:1305.5786 , equally check out the series Emergent algebras as combinatory logic part I, part II, part III, part IV, ). This is not an algebraic theory, but a formalism which contains lambda calculus.
The first figure describes a dictionary of objects which appear in these three formalisms. In the first column you find objects as they appear in dilation structures – emergent algebra formalism. In the second column you find the corresponding object in the binary trees formalism. In the third column there are the respective objects as they appear in the emergent algebra sector of the graphic lambda calculus.
Some comments: the first two rows are about an object called
- “dilation (of coefficient , with , a commutative group)” in dilation structures, which is a operation in emergent algebras, indexed by , (the second row is about dilations of coefficient ),
- it is an elementary binary tree with the node decorated by white (for ) or black (for )
- it is one of the elementary gates in graphic lambda calculus.
The third row is about the “approximate sum” in dilation structures, which is a composite operation in emergent algebras, which is a certain graph in graphic lambda calculus.
The fourth row is about the “approximate difference” and the fifth about the “approximate inverse”.
What is different between these rows?
- In the first row we have an algebra structure based on identities between composites of operations defined on a set.
- In the second row we have trees with leaves decorated by labels from an alphabet (of formal variables) or terms constructed recursively from those .
- In the third row we have graphs with no variable names. (Recall that in graphic lambda calculus there are no variable names. Everything written in red can be safely erased, it is put there only for the convenience of the reader.)
Let’s see now the dictionary of identities/moves.
The most important comment is that identities in emergent algebras become moves in the other two formalisms. A succession of moves is in fact a proof for an identity.
The names of the identities or moves have been commented in many places, you see there names like “Reidemeister move” which show relations to knot diagrams, etc. See this post for the names of the moves and relations to knot diagrams, as well as section 6 from arXiv:1305.5786 .
Let’s read the first column: it says that from an algebraic viewpoint an emergent algebra is a one parameter family (indexed by ) of idempotent right quasigroups. From the geometric point of view of dilation structures, it is a formalisation of properties expected from an object called “dilation”, namely that it preserves the base-point ( “” in the figure), that a composition of dilations of coefficients , with the same base-point, is again a dilation, of coefficient , etc.
In the second column we see two moves, R1 and R2, which can be applied anywhere in a decorated binary tree, as indicated.
In the third column we see that these moves are among the moves from graphic lambda calculus, namely that R1 is in fact related to the oriented Reidemeister move R1a, so it has the same name.
The fact that the idempotent right quasigroup indexed by the neutral element of , denoted by , is trivial, has no correspondent for binary trees, but it appears as the move (ext 2) in graphic lambda calculus. Through the intermediary of this move appears the univalent termination gate.
These are the common moves. To these moves add, for the part of the emergent algebra sector, the R1b move, the local fan-out moves and some pruning moves. There is also the global fan-out move which is needed, but we are going to replace it by a local move which has the funny name of “linearity of fan-out”, but that’s for later.
The local fan-out moves and the pruning moves are needed for the emergent algebra sector but not for the binary trees or emergent algebras. They are the price we have to pay for eliminating variable names. See the algorithm for producing graphs from lambda calculus terms, for what concerns their use for solving the same problem for untyped lambda calculus. (However, the emergent algebra sector is to be compared not with the lambda calculus sector, but with the combinatory logic sector, more about this in a further post.)
We don’t need all pruning moves, but only one which, together with the local fan-out moves, form a family which could be aptly called:
(notice I consider a reversible local pruning)
Grouping moves like this makes a nice symmetry with the fact that is a commutative group, as remarked here.
As concerns the R1b move, which is the one from the next figure, I shall use it only if really needed (for the moment I don’t). It is needed for the knot diagrams made by emergent algebra gates sector, but it is not yet clear to me if we need it for the emergent algebra sector.
However, there is a correspondent of this move for emergent algebras. Indeed, recall that a right quasigroup is a a quasigroup if the equation has a solution, which is unique. If our emergent algebra is in fact a (family of) quasigroup(s) , as happens for the cases of conical groups or for symmetric spaces in the sense of Loos (explained in arXiv:1005.5031 ), then in particular it follows that the equation has only the solution (for ). This last statement has the R1b move as a correspondent in the realm of the emergent algebra sector.
Until now we have only local moves in the emergent algebra sector. We shall see that we need a global move (the global fan-out) in order to prove that the dictionary works, i.e. for proving the fundamental identities of emergent algebras within the graphic lambda calculus formalism. The goal will be to replace the global fan-out move by a new local move (i.e. one which is not a consequence of the existing moves of graphic lambda calculus). This new move will turn out to be a familiar sight, because it is related to the way we see linearity in emergent algebras.
I continue from Parallel transport in spaces with dilations, I. Recall that we have a set , which could be see as the complete directed graph . By a construction using binary decorated trees, with leaves in , we obtain first a set of finite trees , then we put an equivalence relation on this set, namely two finite trees and are close if is a finite tree. The class of finite points is formed by the equivalence classes of finite trees with respect to the closeness relation .
Notice that the equality relation is , in this world. This equality relation is generated by the “oriented Reidemeister moves” R1a and R2a, which appear also as moves in graphic lambda calculus. (By the way, this construction can be made in graphic lambda calculus, which has the moves R1a and R2a. In this way we obtain a higher level of abstraction, because in the process we eliminate the set . Graphic lambda calculus does not need variables. More about this at a future time.) If you are not comfortable with this equality relation than you can just factorize with it and replace it by equality.
It is clear that to any “point” is associated a finite point . Immediate questions jump into the mind:
- (Q1) Is the function injective? Otherwise said, can you prove that if then is not a finite tree?
- (Q2) What is the cardinality of ? Say, if is finite is then infinite ?
Along with these questions, a third one is almost immediate. To any two finite trees and is associated the function defined by
The function is well defined: for any we have , by definition. Therefore , because .
Consider now the groupoid with the set of objects and the set of arrows generated by the arrows from to . The third question is:
- (Q3) What is the isotropy group of a finite point (in particular ) in this groupoid? Call this isotropy group and remark that because the groupoid is connected, it follows that the isotropy groupoid does not depend on the object (finite point), in particular is the same at any point (seen of course as ).
In a future post I shall explain the answers to these questions, which I think they are the following:
- Q1: yes.
- Q2: infinite.
- Q3: a kind of free nilpotent group.