Gromov’s Ergobrain

Misha Gromov updated his very interesting “ergobrain” paper

Structures, Learning and Ergosystems: Chapters 1-4, 6

Two quotes I liked: (my emphasis)

The concept of the distance between, say, two locations on Earth looks simple enough, you do not think you need a mathematician to tell you what distance is. However, if you try to explain what you think you understand so well to a computer program you will be stuck at every step.  (page 76)

Our ergosystems will have no explicit knowledge of numbers, except may be for a few small ones, say two, three and four. On the contrary, neurobrains, being physical systems, are run by numbers which is reflected in their models, such as neural networks which sequentially compose addition of numbers with functions in one variable.

An unrestricted addition is the essential feature of “physical numbers”, such as mass, energy, entropy, electric charge. For example, if you bring together $\displaystyle 10^{30}$  atoms, then, amazingly, their masses add up […]

Our ergosytems will lack this ability. Definitely, they would be bored to death if they had to add one number to another $\displaystyle 10^{30}$ times. But the $\displaystyle 10^{30}$ -addition, you may object, can be implemented by $\displaystyle log_{2} 10^{30} \sim 100$ additions with a use of binary bracketing; yet, the latter is a non-trivial structure in its own right that our systems, a priori, do not have.  Besides, sequentially performing even 10 additions is boring. (It is unclear how Nature performs “physical addition” without being bored in the process.) (page 84)

Where is this going? I look forward to learn.

Two papers on arXiv

I put on arxiv two papers

The paper Computing with space contains too may ideas, is too dense, therefore much of it will not be read, as I was warned repeatedly. This is the reason to do again what I did with Introduction to metric spaces with dilations, which is a slightly edited part of the paper A characterization of sub-riemannian spaces as length dilation structures. Apparently the part (Introduction to ..), the small detail,  is much more read  than the whole (A characterization…).

Concerning the second paper “Normed groupoids…”, it is an improvement of the older paper. Why did I not updated the older paper? Because I need help, I just don’t understand where this is going (and why such direction of research was not explored before).

Escape property of the Gleason metric and sub-riemannian distances again

The last post of Tao from his series of posts on the Hilbert’s fifth problem contains interesting results which can be used for understanding the differences between Gleason distances and sub-riemannian distances or, more general, norms on groups with dilations.

For normed groups with dilations see my previous post (where links to articles are also provided). Check my homepage for more details (finally I am online again).

There is also another post of mine on the Gleason metric (distance) and the CC (or sub-riemannian) distance, where I explain why the commutator estimate (definition 3, relation (2) from the last post of Tao) forces “commutativity”, in the sense that a sub-riemannian left invariant distance on a Lie group which has the commutator estimate must be a riemannian distance.

What about the escape property (Definition 3, relation (1) from the post of Tao)?

From his Proposition 10 we see that the escape property implies the commutator estimate, therefore a sub-riemannian left invariant distance with the escape property must be riemannian.

An explanation of this phenomenon can be deduced by using the notion of “coherent projection”, section 9 of the paper

A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111

in the very particular case of sub-riemannian Lie groups (or for that matter normed groups with dilations).

Suppose we have a normed group with dilations $(G, \delta)$ which has another left invariant dilation structure on it (in the paper this is denoted by a “ $\delta$ bar”, here I shall use the notation $\alpha$ for this supplementary dilation structure).

There is one such a dilation structure available for any Lie group (notice that I am not trying to give a proof of the H5 problem), namely for any $\varepsilon > 0$ (but not too big) $\alpha_{\varepsilon} g = \exp ( \varepsilon \log (g))$

(maybe interesting: which famous lemma is equivalent with the fact that $(G,\alpha)$ is a group with dilations?)
Take $\delta$ to be a dilation structure coming from a left-invariant distribution on the group . Then $\delta$ commutes with $\alpha$ and moreover

(*) $\lim_{\varepsilon \rightarrow 0} \alpha_{\varepsilon}^{-1} \delta_{\varepsilon} x = Q(x)$

where $Q$ is a projection: $Q(Q(x)) = x$ for any $x \in G$.

It is straightforward to check that (the left-translation of) $Q$ (over the whole group) is a coherent projection, more precisely it is the projection on the distribution!

Exercise: denote by $\varepsilon = 1/n$ and use (*) to prove that the escape property of Tao implies that $Q$ is (locally) injective. This implies in turn that $Q = id$, therefore the distribution is the tangent bundle, therefore the distance is riemannian!

UPDATE:    See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for understanding in detail the role of the escape property in the proof of the Hilbert 5th problem.

Planar rooted trees and Baker-Campbell-Hausdorff formula

Today on arXiv was posted the paper

with the abstract

We introduce the combinatorial notion of posetted trees and we use it in order to write an explicit expression of the Baker-Campbell-Hausdorff formula.

The paper may be relevant (check also the bibliography!) for the subject of writing “finitary“, “noncommutative” BCH formulae, from self-similarity arguments using dilations.

Gleason metric and CC distance

In the series of posts on Hilbert’s fifth problem, Terence Tao defines a Gleason metric, definition 4 here, which is a very important ingredient of the proof of the solution to H5 problem.

Here is Remark 1. from the post:

The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

I want to explain why this is true. Look at the proof of theorem 7. The problem comes from the commutator estimate (1). I shall reproduce the relevant part of the proof because I don’t yet know how to write good-looking latex posts:

From the commutator estimate (1) and the triangle inequality we also obtain a conjugation estimate $\displaystyle \| ghg^{-1} \| \sim \|h\|$

whenever ${\|g\|, \|h\| \leq \epsilon}$. Since left-invariance gives $\displaystyle d(g,h) = \| g^{-1} h \|$

we then conclude an approximate right invariance $\displaystyle d(gk,hk) \sim d(g,h)$

whenever ${\|g\|, \|h\|, \|k\| \leq \epsilon}$.

The conclusion is that the right translations in the group are Lipschitz (with respect to the Gleason metric). Because this distance (I use “distance” instead of “metric”) is also left invariant, it follows that left and right translations are Lipschitz.

Let now G be a connected Lie group with a left-invariant distribution, obtained by left translates of a vector space D included in the Lie algebra of G. The distribution is completely non-integrable if D generates the Lie algebra by using the + and Lie bracket operations. We put an euclidean norm on D and we get a CC distance on the group defined by: the CC distance between two elements of the group equals the infimum of lengths of horizontal (a.e. derivable, with the tangent in the distribution) curves joining the said points.

The remark 1 of Tao is a consequence of the following fact: if the CC distance is right invariant then D equals the Lie algebra of the group, therefore the distance is riemannian.

Here is why: in a sub-riemannian group (that is a group with a distribution and CC distance as explained previously) the left translations are Lipschitz (they are isometries) but not all right translations are Lipschitz, unless D equals the Lie algebra of G. Indeed, let us suppose that all right translations are Lipschitz. Then, by Margulis-Mostow version (see also this) of the Rademacher theorem , the right translation by an element “a” is Pansu derivable almost everywhere. It follows that the Pansu derivative of the right translation by “a” (in almost every point) preserves the distribution. A simple calculus based on invariance (truly, some explanations are needed here) shows that by consequence the adjoint action of “a” preserves D. Because “a” is arbitrary, this implies that D is an ideal of the Lie algebra. But D generates the Lie algebra, therefore D equals the Lie algebra of G.

If you know a shorter proof please let me know.

UPDATE: See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for details of the role of the Gleason metric  in the proof of the Hilbert 5th problem.

Curvature and Brunn-Minkowski inequality

A beautiful paper by Yann Ollivier and Cedric Villani

A curved BRUNN–MINKOWSKI INEQUALITY on the discrete hypercube OR: WHAT IS THE RICCI CURVATURE OF THE DISCRETE  HYPERCUBE?

The Brunn-Minkowski inequality  says that  the log  of the volume (in euclidean spaces) is concave. The concavity inequality is improved, in riemannian manifolds with Ricci curvature at least K, by a quadratic term with coefficient proportional with K.

The paper is remarkable in many ways. In particular are compared two roads towards curvature in spaces more general than riemannian: the coarse curvature introduced by Ollivier and the other based on the displacement convexity of the entropy function (Felix Otto , Cedric Villani, John Lott, Karl-Theodor Sturm), studied by many researchers. Both are related to  Wasserstein distances . NONE works for sub-riemannian spaces, which is very very interesting.

In few words, here is the description of the coarse Ricci curvature: take an epsilon and consider the application from the metric space (riemannian manifold, say) to the space of probabilities which associates to a point from the metric space the restriction of the volume measure on the epsilon-ball centered in that point (normalized to give a probability). If this application is Lipschitz with constant L(epsilon) (on the space of probabilities take the L^1 Wassertein distance) then the epsilon-coarse Ricci curvature times epsilon square is equal to 1 minus L(epsilon) (thus we get a lower bound of the Ricci curvature function, if we are in a Riemannian manifold). Same definition works in a discrete space (this time epsilon is fixed).
The second definition of Ricci curvature comes from inverse engineering of the displacement convexity inequality discovered in many particular spaces. The downside of this definition is that is hard to “compute” it.

Initially, this second definition was related to the L^2 Wasserstein distance which,  according to Otto calculus, gives to the space of probabilities (in the L^2 frame) a structure of an infinite dimensional riemannian manifold.

Concerning the sub-riemannian spaces, in the first definition the said application cannot be Lipschitz and in the second definition there is (I think) a manifestation of the fact that we cannot put, in a metrically acceptable way, a sub-riemannian space into a riemannian-like one, even infinite dimensional.