# Uniform spaces, coarse spaces, dilation spaces

This post is related to the previous What’s the difference between uniform and coarse structures? (I).  However, it’s not part (II).

Psychological motivation.  Helge Gloecker made a clear mathscinet review of my paper Braided spaces with dilations and sub-riemannian symmetric spaces.  This part of the review motivated me to start writing some more about dilatation structures, some of it explained in the next section of this post. Here is the motivating part of the review:

“Aiming to put sub-Riemannian geometry on a purely metric footing (without recourse to differentiable structures), the author developed the concept of a dilatation structure and related notions in earlier works [see J. Gen. Lie Theory Appl. 1 (2007), no. 2, 65–95; MR2300069 (2008d:20074); Houston J. Math. 36 (2010), no. 1, 91–136; MR2610783 (2011h:53032); Commun. Math. Anal. 11 (2011), no. 2, 70–111; MR2780883 (2012e:53051)].

[…]

In a recent preprint [see “Emergent algebras”, arXiv:0907.1520], the author showed that idempotent right quasigroups are a useful auxiliary notion for the introduction (and applications) of dilatation structures.

[…]

The author sketches how dilatation structures can be defined with the help of uniform irqs, suppressing the technical and delicate axioms concerning the domains of the locally defined maps. More precisely, because the maps are only locally defined, the structures which occur are not uniform irqs at face value, but only suitable analogues using locally defined maps (a technical complication which is not mentioned in the article).”

This was done before, several times, for dilatation structures, but Helge is right that it has not been done for emergent algebras.

Which brings me to the main point: I started a paper with the same (but temporary) name as this post.  The paper is based on the following ideas.

Fields of dilations which generate their own topology, uniformity, coarse structure, whatever.    In a normed real vector space $(V, \| \cdot \|)$ we may use dilations as replacements of metric balls. Here is the simple idea.

Let us define, for any $x \in V$, the domain of $x$ as $U(x)$,  the ball with center $x$, of radius one, with respect to the distance $d(x,y) = \|x - y \|$. In general, let $B(x, r)$  be the  metric ball with respect to the distance induced by the norm.

Also, for any $x \in V$ and $\varepsilon \in (0,+\infty)$, the dilation based at $x$, with coefficient $\varepsilon$ is the function $\delta^{x}_{\varepsilon} y = x + \varepsilon ( -x + y)$ .

I use these notations to write the  ball of radius $\varepsilon \in (0,1]$ as $B(x,\varepsilon) = \delta^{x}_{\varepsilon} U(x)$ and give it the fancy name dilation ball of coefficient $\varepsilon$.

In fact, spaces with dilations, or dilatation structures, or dilation structures, are various names for spaces endowed with fields of dilations which satisfy certain axioms. Real normed vector spaces are examples of spaces with dilations, as a subclass of the larger class of conical groups with dilations, itself just a subclass of groups with dilations. Regular sub-riemannian manifolds are examples of spaces with dilations without a predefined group structure.

For any space with dilations, then, we may ask what can we get by forgetting the distance function, but keeping the dilation balls.  With the help of those we may define the uniformity associated with a field of dilations and then ask that the field of dilations is behaves uniformly, and so on.

Another structure, as interesting as the uniformity structure, is the (bounded metric) coarse structure of the space, which  again could be expressed in terms of fields of dilations. As coarse structures and uniform structures are very much alike (only that one is interesting for the small scale, other for the large scale), is there a notion of dilation structure which is appropriate for coarse structures?

I shall be back on this, with details, but the overall idea is that a field of dilations (a notion even weaker than an emergent algebra) is somehow midway between a uniformity structure and a coarse structure.

UPDATE (28.10.2012): I just found bits of the work of Protasov, who introduced the notion of “ball structure” and “ballean”, which is clearly related with the idea of keeping the balls, not the distance. Unfortunately, for the moment I am unable to find a way to read his two monographs on the subject.

# What’s the difference between uniform and coarse structures? (I)

See this, if you need: uniform structure and coarse structure.

Given a set $X$, endowed with an uniform structure or with a coarse structure, Instead of working with subsets of $X^{2}$, I prefer to think about the trivial groupoid $X \times X$.

Groupoids. Recall that a groupoid is a small category with all arrows invertible. We may define a groupoid in term of it’s collection of arrows, as follows: a set $G$ endowed with a partially defined operation $m: G^{(2)} \subset G \times G \rightarrow G$, $m(g,h) = gh$, and an inverse operation $inv: G \rightarrow G$, $inv(g) = g^{-1}$, satisfying some well known properties (associativity, inverse, identity).

The set $Ob(G)$ is the set of objects of the groupoid, $\alpha , \omega: G \rightarrow Ob(G)$ are the source and target functions, defined as:

– the source fonction is $\alpha(g) = g^{-1} g$,

– the target function is $\omega(g) = g g^{-1}$,

– the set of objects is $Ob(G)$ consists of all elements of $G$ with the form $g g^{-1}$.

Trivial groupoid. An example of a groupoid is $G = X \times X$, the trivial groupoid of the set $X$. The operations , source, target and objects are

$(x,y) (y,z) = (x,z)$, $(x,y)^{-1} = (y,x)$

$\alpha(x,y) = (y,y)$, $\omega(x,y) = (x,x)$,

– objects $(x,x)$ for all $x \in X$.

In terms of groupoids, here is the definition of an uniform structure.

Definition 1. (groupoid with uniform structure) Let $G$ be a groupoid. An uniformity $\Phi$ on $G$ is a set $\Phi \subset 2^{G}$ such that:

1. for any $U \in \Phi$ we have $Ob(G) \subset U$,

2. if $U \in \Phi$ and $U \subset V \subset G$ then $V \in \Phi$,

3. if $U,V \in \Phi$ then $U \cap V \in \Phi$,

4. for any $U \in \Phi$ there is $V \in \Phi$ such that $VV \subset U$,  where $VV$ is the set of all $gh$ with $g, h \in V$ and moreover $(g,h) \in G^{(2)}$,

5. if $U \in \Phi$ then $U^{-1} \in \Phi$.

Here is a definition for a coarse structure on a groupoid (adapted from the usual one, seen on the trivial groupoid).

Definition 2.(groupoid with coarse structure) Let $G$ be a groupoid. A coarse structure  $\chi$ on $G$ is a set $\chi \subset 2^{G}$ such that:

1′. $Ob(G) \in \chi$,

2′. if $U \in \chi$ and $V \subset U$ then $V \in \chi$,

3′. if $U,V \in \chi$ then $U \cup V \in \chi$,

4′. for any $U \in \chi$ there is $V \in \chi$ such that $UU \subset V$,  where $UU$ is the set of all $gh$ with $g, h \in U$ and moreover $(g,h) \in G^{(2)}$,

5′. if $U \in \chi$ then $U^{-1} \in \chi$.

Look pretty much the same, right? But they are not, we shall see the difference when we take into consideration how we generate such structures.

Question. Have you seen before such structures defined like this, on groupoids? It seems trivial to do so, but I cannot find this anywhere (maybe because I am ignorant).

UPDATE (01.08.2012): Just found this interesting paper  Coarse structures on groups, by Andrew Nicas and David Rosenthal, where they already did on groups something related to what I want to explain on groupoids. I shall be back on this.

# Two halves of beta, two halves of chora

In this post I want to emphasize a strange similarity between the beta rule in lambda calculus and the chora construction (i.e. encircling a tangle diagram).

Motivation? Even if now clearer, I am still not completely satisfied by the degree of interaction between lambda calculus and emergent algebras, in the proposed lambda-Scale calculus. I am not sure if this is because lambda-Scale is yet not explored, or because there exist a more streamlined version of lambda calculus as a macro over the emergent algebras.

Also, I am working again on the paper put on preview (version 05.06.2012) about planar trivalent graphs ans lambda calculus, after finishing the  course notes on intrinsic sub-riemannian geometry.

So, I let my mind hovering over …

As explained in the draft paper, the beta rule in lambda calculus  is a LOCAL rule, described in this picture (advertised here):

It is made by two halves: the left half contains the lambda abstraction, the right half contains the application operation. In between there is a wire. The rule says that these two halves annihilate somehow and the wire is replaced by a dumb crossing with no information about who’s on top.

Let us contemplate an elementary chora, made also by two halves:

We can associate to this figure a move, which consists in the annihilation of the left (difference gate) and right (sum gate) halves, followed by the replacement of the “wire” by an equivalent crossing

# Preview of two papers, thanks for comments

Here are two papers:

Local and global moves on planary trivalent graphs, lambda calculus and lambda-Scale (update 03.07.2012, final version, appears as arXiv:1207.0332)

Sub-riemannian geometry from intrinsic viewpoint    (update 14.06.2012: final version, appears as arxiv:1206.3093)

which are still subject to change.  Nevertheless most of what I am trying to communicate is there. I would appreciate  mathematical comments.

This is an experiment,  to see what happens if I make previews of papers available, like a kind of a blog of papers in the making.

# Intrinsic characterizations of riemannian and sub-riemannian spaces (I)

In this post I explain what is the problem of intrinsic characterization of riemannian manifolds, in what sense has been solved in full generality by Nikolaev, then I shall comment on the proof of the Hilbert’s fifth problem by Tao.

In the next post there will be then some comments about Gromov’s problem of giving an intrinsic characterization of sub-riemannian manifolds, in what sense I solved this problem by adding a bit of algebra to it. Finally, I shall return to the characterization of riemannian manifolds, seen as particular sub-riemannian manifolds, and comment on the differences between this characterization and Nikolaev’ one.

1. History of the problem for riemannian manifolds. The problem of giving an intrinsic characterization of riemannian manifolds is a classic and fertile one.

Problem: give a metric description of a Riemannian manifold.

Background: A complete riemannian manifold is a length metric space (or geodesic, or intrinsic metric space) by Hopf-Rinow theorem. The problem asks for the recovery of the manifold structure from the distance function (associated to the length functional).

For 2-dim riemannian manifolds the problem has been solved by A. Wald [Begrundung einer koordinatenlosen Differentialgeometrie der Flachen, Erg. Math. Colloq. 7 (1936), 24-46] (“Begrundung” with umlaut u, “Flachen” with umlaut a, sorry for this).

In 1948 A.D. Alexandrov [Intrinsic geometry of convex surfaces, various editions] introduces its 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 Holder smoothness). Many other results deserve to be mentioned (by Reshetnyak, for example).

2. Solution of the problem by Nikolaev. In 1998 I.G. Nikolaev [A metric characterization of riemannian spaces, Siberian Adv. Math. , 9 (1999), 1-58] solves the general problem of intrinsic characterization of $C^{m,\alpha}$ riemannian spaces:

every locally compact length metric space $M$, not linear at one of its points,  with $\alpha$ Holder continuous metric sectional curvature of the “generalized tangent bundle” $T^{m}(M)$ (for some $m=1,2,…$, which admits local geodesic extendability, is isometric to a $C^{m+2}$ smooth riemannian manifold..

Therefore:

• he defines a generalized tangent bundle in metric sense
• he defines a notion of sectional curvature
• he asks some metric smoothness of this curvature

and he gets the result.

3. Gleason metrics and Hilbert’s fifth problem. Let us compare this with the formulation of the solution of the Hilbert’s fifth problem by Terence Tao. THe problem is somehow similar, namely recover the differential structure of a Lie group from its algebraic structure. This time the “intrinsic” object is the group operation, not the distance, as previously.

Tao shows that the proof of the solution may be formulated in metric terms. Namely, he introduces a Gleason metric (definition 4 in the linked post), which will turn to be a left invariant riemannian metric on the (topological) group. I shall not insist on this, instead read the post of Tao and also, for the riemannian metric description, read this previous post by me.

# Lambda-Scale is the new name

The paper on the calculus adapted to emergent algebras has been almost completely rewritten. I submitted it to arXiv, it shall appear tomorrow.

The new name of the paper is “$\lambda$-Scale, a lambda calculus for spaces with dilations”, and it is already available from my page at this link.

# Rules of lambda epsilon calculus

I updated the paper on lambda epsilon calculus.  See the link to the actual version (updated daily) or check the arxiv article, which will be updated as soon as a stable version will emerge.

Here are the rules of this calculus:

(beta *)   $(x \lambda A) \varepsilon B= (y \lambda (A[x:=B])) \varepsilon B$ for any fresh variable $y$,

(R1) (Reidemeister one) if $x \not \in FV(A)$ then $(x \lambda A) \varepsilon A = A$

(R2) (Reidemeister two) if $x \not \in FV(B)$ then $(x \lambda ( B \mu x)) \varepsilon A = B (\varepsilon \mu ) A$

(ext1) (extensionality one)  if  $x \not \in FV(A)$ then $x \lambda (A 1 x) = A$

(ext2) (extensionality two) if  $x \not \in FV(B)$ then $(x \lambda B) 1 A = B$

These are taken together with usual substitution and $\alpha$-conversion.

The relation between the operations from $\lambda \varepsilon$ calculus and emergent algebras is illustrated in the next figure.