# Theatron as an eye

I want to understand what “computing with space” might be. By making  a parallel with the usual computation, there are three ingredients which need to be identified: what are the computing with space equivalents of

1. the universal computing gate (in usual computing this is the transistor)

2. the universal machine (in usual computing this is the Turing machine)

3. what is the universal machine doing by using its arrangement of universal computing gates (in usual computing this is the algorithm).

I think that (3) is (an abstraction of) the activity of map making, or space exploration. The result of this activity is coded by a dilation structure, but I have no idea HOW such a result is achieved. Once obtained though, a mathematical model of the space is the consequence of  a priori assumptions (that we can repeat in principle indefinitely the map making operations) which lead to the emergent algebraic and differential structure of the space.

The universal gate (1), I think, is the dilation gate, or the map-territory relation.

Today I want to pave the way to the discovery of the universal machine (2). This is related to my previous posts The Cartesian Theater: philosophy of mind versus aerography and Towards aerography, or how space is shaped to comply with the perceptions of the homunculus.

My take is that the Greek Theater, or Theatron (as opposed to the “theater in a box”, or Cartesian Theater) is a good model for an universal machine.

For today, I just want to point to the similarities between the theatron and the eye.

The following picture represents the main parts of the theatron (the ancient greek meaning of “theatron” is “place of seeing). In black are written the names of the theatron parts and in red you see the names of the corresponding parts of the eye, according to the proposed similarity.

Let me proceed with the meaning of these words:

– Analemmata means the pedestal of a sundial (related with analemma and analemmatic sundial; basically a theatron is an analemmatic sundial, with the chorus as the gnomon). I suggest to parallel this with the choroid of the eye.

– Diazomata (diazoma means “belt”), proposed to be similar with the retina.

Prohedria (front seating) is a privilege to sit in the first few rows at the bottom of the viewing area. Similar with the fovea (small pit), responsible for sharp central vision.

Skene (tent), the stage building, meant to HIDE the workings  of the actors which are not part of the show, as well as the masks and other materials. When a character dies, it happens behind the skene. Eventually, the skene killed the chorus and  became the stage. The eye equivalent  of this is the iris.

Parodos (para – besides, counter, and ode – song) entrance of the chorus. Eye equivalent is the crystalline lens.

– Orchestra, the ancient greek stage, is the place where the chorus acts, the center of the greek theater. Here we pass to abstraction: the eye correspondent is the visual field.

# Approximate algebraic structures, emergent algebras

I updated and submitted for publication the paper “Emergent algebras“.

This is the first paper where emergent algebras appear. The subject is further developed in the paper “Braided spaces with dilations and sub-riemannian symmetric spaces“.

I strongly believe this is a very important notion, because it shows how  both the differential and algebraic realms  emerge naturally, from abstract nonsense. It is a “low tech” approach, meaning that I don’t use in the construction any “high tech” mathematical object, everything is growing from the grass.

One interesting fact, apart from the strange ideas of the paper (it is already verified that reading the paper algebraists will not understand easily the strength of the axiom concerning uniform convergence and analysts will not care enough about the occurrence of algebraic structure very much alike quandles and racks), is that an emergent algebra can also be seen as an approximate algebraic structure! But in a different sense than approximate groups.  The operations themselves are approximately associative, for example.

And my next question is: is this a really different notion of approximate algebraic structure than approximate groups? Or there is a way to see, for example, an approximate group (btw, why not an approximate symmetric space in the sense of Loos, whatever this could mean?) as an emergent algebra?

My hope is that the answer is YES.

UPDATE:   No, in fact there are reasons to think that there is a complementarity, there is a mathematical object standing over both, which may be called POSITIONAL SYSTEM, more soon, but see also this previous post of mine.

Here is the abstract of the paper:

“Inspired from research subjects in sub-riemannian geometry and metric geometry, we propose uniform idempotent right quasigroups and emergent algebras as an alternative to differentiable algebras.
Idempotent right quasigroups (irqs) are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). To any uniform idempotent right quasigroup can be associated an approximate differential calculus, with Pansu differential calculus in sub-riemannian geometry as an example.
An emergent algebra A over a uniform idempotent right quasigroup X is a collection of operations such that each operation emerges from X, meaning that it can be realized as a combination of the operations of the uniform irq X, possibly by taking limits which are uniform with respect to a set of parameters.
Two applications are considered: we prove a bijection between contractible groups and distributive uniform irqs (uniform quandles) and that some symmetric spaces in the sense of Loos may be seen as uniform quasigroups with a distributivity property. “

# On the difference of two Lipschitz functions defined on a Carnot group

Motivation for this post: the paper “Lipschitz and biLipschitz Maps on Carnot Groups” by William Meyerson. I don’t get it, even after several readings of the paper.

The proof of Fact 2.10 (page 10) starts by the statement that the difference of two Lipschitz functions is Lipschitz and the difference of two Pansu differentiable functions is differentiable.

Let us see: we have a Carnot group (which I shall assume is not commutative!) $G$ and two functions $f,g: U \subset G \rightarrow G$, where $U$ is an open set in $G$. (We may consider instead two Carnot groups $G$ and $H$ (both non commutative) and two functions $f,g: U \subset G \rightarrow H$.)

Denote by $h$ the difference of these functions: for any $x \in U$ $h(x) = f(x) (g(x))^{-1}$  (here the group operations  and inverses are denoted multiplicatively, thus if $G = \mathbb{R}^{n}$ then $h(x) = f(x) - g(x)$; but I shall suppose further that we work only in groups which are NOT commutative).

1.  Suppose $f$ and $g$ are Lipschitz with respect to the respective  CC left invariant distances (constructed from a choice of  euclidean norms on their respective left invariant distributions).   Is the function $h$ Lipschitz?

NO! Indeed, consider the Lipschitz functions $f(x) = x$, the identity function,  and $g(x) = u$ a constant function, with $u$ not in the center of $G$. Then $h$ is a right translation, notoriously NOT Lipschitz with respect to a CC left invariant distance.

2. Suppose instead that $f$ and $g$ are everywhere Pansu differentiable and let us compute the Pansu “finite difference”:

$(D_{\varepsilon} h )(x,u) = \delta_{\varepsilon^{-1}} ( h(x)^{-1} h(x \delta_{\varepsilon} u) )$

We get that $(D_{\varepsilon} h )(x,u)$ is the product w.r.t. the group operation of two terms: the first is the conjugation of the finite difference $(D_{\varepsilon} f )(x,u)$  by $\delta_{\varepsilon^{-1}} ( g(x) )$ and the second term is the finite difference   $(D_{\varepsilon} g^{-1} )(x,u)$.  (Here  $Inn(u)(v) = u v u^{-1}$ is the conjugation of $v$ by $u$ in the group $G$.)

Due to the non commutativity of the group operation, there should be some miracle in order for the finite difference of $h$ to converge, as $\varepsilon$ goes to zero.

We may take instead the sum of two differentiable functions, is it differentiable (in the sense of Pansu?). No, except in very particular situations,  because we can’t get rid of the conjugation, because the conjugation is not a Pansu differentiable function.

# Non-euclidean analysis of dilation structures

Here are  the slides for the following talk:

Non-euclidean analysis of dilation structures

This is a general audience (of mathematicians) talk, a kind of appetizer for the subject.

# Non-Euclidean analysis, a bit of recent history

Being an admirer of bold geometers who discovered that there is more to geometry than euclidean geometry, I believe that the same is true for analysis. In my first published paper “The topological substratum of the derivative” (here is a scan of this hard to find paper), back in 1993, I advanced the idea that there are as many “analyses” as the possible fields of dilations, but I was not aware about Pierre Pansu huge paper from 1989 “Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un” (sorry for the missing accents, I am still puzzled by the keyboard of the computer I am using to write this post), where he invents what is now called “Pansu calculus”, which is the analysis associated to a Carnot group.

The same idea is then explored in the papers “Sub-riemannian geometry and Lie groups, I“, “Tangent bundles to sub-riemannian groups“, “Sub-riemannian geometry and Lie groups II“. These papers have not been published (only put on arXiv), because at that moment I hoped that the www will change publishing quick (I still do believe this, but now I am just a bit wiser, or forced by bureaucracy to publish or perish), so one could communicate not only the very myopic technical, incremental result, but also the ideas behind, the powerful  meme.

During those years (2001-2005) I have been in Lausanne, trying to propagate the meme around, in Europe, as I said previously. There were mixed results, people were not taking this serious enough, according to my taste. Sergey Vodopyanov had ideas which were close to mine, except that he was trying to rely on what I call “euclidean analysis”, instead of intrinsic techniques, as witnessed by his outstanding results concerning detailed proofs in low-regularity sub-riemannian geometry. (I was against such results by principle, because what is $C^{1,1}$ but euclidean regularity? but the underlying ideas were very close indeed).

In a very naive way I tried to propagate the meme further, by asking for a visit at IHES, in 2004, when I had the pleasure to meet Pierre Pansu and Andre Bellaiche, then I dared to ask for another visit immediately and submitted the project

“Non-Euclidean Analysis” start-up

which I invite you to read. (The project was rejected, for good reasons, I was already there visiting and suddenly I was asking for another, much longer visit)

Then, from 2006 I went back to basics and proposed axioms for this, that is how dilation structures appeared (even if the name and a definition containing the most difficult axiom was already proposed in the previous series of papers on sub-riemannian geometry and Lie groups.  See my homepage for further details and papers (published this time).

I see now that, at least at the level of names of grant projects, the meme is starting to spread. Here is the “Sub-riemannian geometric analysis in Lie groups” GALA project and here is the more recent “Geometric measure theory in Non Euclidean spaces” GeMeThNES project.

# Bipotentials, variational formulations

The paper on the use of bipotentials in variational formulations is finally submitted, also available on arxiv here. See also this presentation.

In case you wonder how this could be related with other subjects commented on this blog, then wait to see “A gallery of emergent algebras”, where it shall be explained the connection between convex analysis and an emergent algebra related to a semidirect product between the semigroup of words over a symplectic space and $\mathbb{R}$.

# Proof mining and approximate groups, announcement

The last weeks have been very busy for personal reasons. I shall come back to writing on this blog in short time.

With Laurentiu Leustean we won (a month ago, but the project has been submitted this Spring) the financing for our research project

“Proof mining in metric anaysis, geometric group theory and ergodic theory”

(project PN-II-ID-PCE-2011-3-0383). Laurentiu is a specialist in proof mining with applications to geodesic spaces and ergodic theory; I am interested in emergent algebras, particularly in dilation structures, so one of our aims (in this project) is to understand why nilpotent like structures appear in the famous Gromov theorem on groups of polynomial growth, as well in approximate groups, by using proof mining techniques for “finitizing” emergent algebras, roughly.

This program is very close to one of the programs of Terence Tao, who continues his outstanding research on approximate groups. The following post

Ultraproducts as a bridge between hard analysis and soft analysis

made me happy because it looks like confirming that our dreams (for the moment) have a correspondent in reality and probably ideas like this are floating in the air.

UPDATE 25.10: Today a new post of Tao announces the submission on arxiv of the paper by him, Emmanuel Breuillard and Ben Green, “The structure of approximate groups“. I look forward to study it, to see if they explain why nilpotent structures appear in the limit. My explanation, in a different context, “A characterization of sub-riemannian spaces…”, related also to work of Gromov, namely why in sub-riemannian geometry nilpotent groups appear as models of metric tangent spaces, is that this is a feature of an emergent algebra. See also previous posts, like Principles: randomness/structure or emergent from a common cause?.