Kashiwara-Vergne and the noncommutative BCH

For further reference, it may be that the problem of the noncommutative BCH is related to, or could benefit from the solution of the KW conjecture given by Alekseev and Torossian. So it is a very hard problem, maybe.

Noncommutative Baker-Campbell-Hausdorff formula: the problem

I come back to a problem alluded in a previous post, where the proof of the Baker-Campbell-Hausdorff formula from this post by Tao is characterized as “commutative”, because of the “radial homogeneity” condition in his Theorem 1 , which forces commutativity.

Now I am going to try to explain this, as well as what the problem of a “noncommutative” BCH formula would be.

Take a Lie group and identify a neighbourhood of its neutral element with a neighbourhood of the element of its Lie algebra. This is standard for Carnot groups (connected, simply connected nilpotent groups which admit a one parameter family of contracting automorphisms), where the exponential is bijective, so the identification is global. The advantage of this identification is that we get rid of log’s and exp’s in formulae.

For every define a deformation of the group operation (which is denoted multiplicatively), by the formula

(1)               

Then we have as .

Denote by the Lie bracket of the (Lie algebra of the) group with initial operation and likewise denote by the Lie bracket of the operation .

The relation between these brackets is: .

From the Baker-Campbell-Hausdorff formula we get:

,

(for reasons which will be clear later, I am not using the commutativity of addition), therefore

(2)                as        .

Remark that (2) looks like a valid definition of the Lie bracket which is not related to the group commutator. Moreover, it is a formula where we differentiate only once, so to say. In the usual derivation of the Lie bracket from the group commutator we have to differentiate twice!

Let us now pass to a slightly different context: suppose is a normed group with dilations (the norm is for simplicity, we can do without; in the case of “usual” Lie groups, taking a norm corresponds to taking a left invariant Riemannian distance on the group).

is a normed group with dilations if

• it is a normed group, that is there is a norm function defined on with values in , denoted by , such that

– iff (the neutral element)

–

–

– “balls” are compact in the topology induced by the distance $d(x,y) = \|x^{-1} y\|$,

• and a “multiplication by positive scalars” with the properties:

– , and as $s \rightarrow 0$; also ,

– define as previously, by the formula (1) (only this time use the multiplication by positive scalars). Then

      as     

uniformly with respect to in an arbitrarry closed ball.

– , uniformly with respect to in a closed ball, and moreover implies .

Comments:

1. In truth, everything is defined in a neighbourhood of the neutral element, also has only to be a local group.

2. the operation is a (local) group operation and the function is a norm for this operation, which is also “homogeneous”, in the sense

.

Also we have the distributivity property , but generally the dot operation is not commutative.

3. A Lie group with a left invariant Riemannian distance and with the usual multiplication by scalars (after making the identification of a neighbourhood of the neutral element with a neighbourhood in the Lie algebra) is an example of a normed group with dilations, with the norm .

4. Any Carnot group can be endowed with a structure of a group with dilations, by defining the multiplication by positive scalars with the help of its intrinsic dilations. Indeed, take for example a Heisenberg group with the operation

multiplication by positive scalars

and norm given by

Then we have , for any and for any .

Carnot groups are therefore just a noncommutative generalization of vector spaces, with the addition operation $+$ replaced by a noncommutative operation!

5. There are many groups with dilations which are not Carnot groups. For example endow any Lie group with a left invariant sub-riemannian structure and hop, this gives a norm group with dilations structure.

In such a group with dilations the “radial homogeneity” condition of Tao implies that the operation is commutative! (see the references given in this previous post). Indeed, this radial homogeneity is equivalent with the following assertion: for any and any

which is called elsewhere “barycentric condition”. This condition is false in any noncommutative Carnot group! What it is true is the following: let, in a Carnot group, be any solution of the equation

for given and $s \in (0,1)$. Then

,

(so the solution is unique) where the sum is taken with respect to the group operation (noncommutative series).

Problem of the noncommutative BCH formula: In a normed group with dilations, express the group operation as a noncommutative series, by using instead of “” the operation “” and by using a definition of the “noncommutative Lie bracket” in the same spirit as (2), that is something related to the asymptotic behaviour of the “approximate bracket”

(3)         .

Notice that there is NO CHANCE to have a limit like the one in (2), so the problem seems hard also from this point of view.

Also notice that if is a Carnot group then

(that is like it is equal to , remember)

which is normal, if we think about as being a kind of noncommutative vector space, even of may be not commutative.

So this noncommutative Lie bracket is not about commutators!

Topological substratum of the derivative

Until recently, on my home page was a link to the scan of the paper

 The topological substratum of the derivative (I), Math. Reports (Stud. Cerc. Mat.) 45, 6,       (1993), 453-465

which is no longer visible now. But maybe it deserves a post here, because is my oldest attempt to understand differential calculus as an abstract matter and to look to new forms of it.

To me it became clear that differential calculus admits variants, in the same spirit as euclidean geometry admitting non-euclidean variants. At that moment I had no really intersting examples of such a “non-euclidean” differential calculus, so I switched to other research subjects. Nobody pointed to me the huge paper

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, The Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

by Pierre Pansu. It was only luck that in 2000, at Lausanne, I met Sergey Vodop’yanov (from Sobolev Institute of Mathematics). He started to explain to me what Carnot groups are and I was thrilled to   learn that examples I needed previously are numerous in sub-riemannian geometry.

With the right frame of mind (at least I think so), that of intrinsic dilations, I  started then to study sub-riemannian geometry.

Planar rooted trees and Baker-Campbell-Hausdorff formula

Today on arXiv was posted the paper

Posetted trees and Baker-Campbell-Hausdorff product, by Donatella Iacono, Marco Manetti

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.

Entering “chora”, the infinitesimal place

There is a whole discussion around the key phrases “The map is not the territory” and “The map is the territory”. From the wiki entry on the map-territory relation, we learn that Korzybski‘s dictum “the map is not the territory” means that:

A) A map may have a structure similar or dissimilar to the structure of the territory,

B) A map is not the territory.

Bateson, in “Form, Substance and Difference” has a different take on this: he starts by explaining the pattern-substance dichotomy

Let us go back to the original statement for which Korzybski is most famous—the statement that the map is not the territory. This statement came out of a very wide range of philosophic thinking, going back to Greece, and wriggling through the history of European thought over the last 2000 years. In this history, there has been a sort of rough dichotomy and often deep controversy. There has been violent enmity and bloodshed. It all starts, I suppose, with the Pythagoreans versus their predecessors, and the argument took the shape of “Do you ask what it’s made of—earth, fire, water, etc.?” Or do you ask, “What is its pattern?” Pythagoras stood for inquiry into pattern rather than inquiry into substance.¹ That controversy has gone through the ages, and the Pythagorean half of it has, until recently, been on the whole the submerged half.

Then he states his point of view:

We say the map is different from the territory. But what is the territory? […] What is on the paper map is a representation of what was in the retinal representation of the man who made the map–and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all.

Always the process of representation will filter it out so that the mental world is only maps of maps of maps, ad infinitum.

At this point Bateson puts a very interesting footnote:

Or we may spell the matter out and say that at every step, as a difference is transformed and propagated along its pathways, the embodiment of the difference before the step is a “territory” of which the embodiment after the step is a “map.” The map-territory relation obtains at every step.

Inspired by Bateson, I want to explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using tangle diagrams.

Let us imagine that the exploration of the territory provides us with an atlas, a collection of maps, mathematically understood as a family of two operations (an “emergent algebra”). We want to organize this spatial information in a graphical form which complies with Bateson’s footnote: map and territory have only local meaning in the graphical representation, being only the left-hand-side (and r-h-s respectively) of the “making map” relation.

Look at the following figure:

In the figure from the left, the “v” which decorates an arc, represents a point in the “territory”, that is the l-h-s of the relation, the “u” represents a “pixel in the map”, that is the r-h-s of a relation. The relation itself is represented by a crossing decorated by an epsilon, the “scale” of the map.

The opposite crossing, see figure from the right, is the inverse relation.

Imagine now a complex diagram, with lots of crossings, decorated by various
scale parameters, and segments decorated with points from a space X which
is seen both as territory (to explore) and map (of it).

In such a diagram the convention map-territory can be only local, around each crossing.

There is though a diagram which could unambiguously serve as a symbol for
“the place (near) the point x, at scale epsilon” :

In this diagram, all crossings which are not decorated have “epsilon” as a decoration, but this decoration can be unambiguously placed near the decoration “x” of the closed arc. Such a diagram will bear the name “infinitesimal place (or chora) x at scale epsilon”.

“Metric spaces with dilations”, the book draft updated

Here is the updated version.

Many things left to be done and to explain properly, like:

• the word tangent bundle and more flexible notions of smoothness, described a bit hermetically and by means of examples here, section 6,
• the relation between curvature and how to perform the Reidemeister 3 move, the story starts here in section 6 (coincidence)
• why the Baker-Campbell-Hausdorff formula can be deduced from self-similarity arguments (showing in particular that there is another interpretation of the Lie bracket than the usual one which says that the bracket is related to the commutator). This will be posted on arxiv soon. UPDATE:  see this post by Tao on the (commutative, say) BCH formula. It is commutative because of his “radial homogeneity” axiom in Theorem 1, which is equivalent with the “barycentric condition” (Af3) in Theorem 2.2 from “Infinitesimal affine geometry…” article.

• a gallery of emergent algebras, in particular you shall see what “spirals” are (a kind of generalization of rings, amazingly connecting by way of an example with another field of interests of mine, convex analysis).

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

whenever . Since left-invariance gives

we then conclude an approximate right invariance

whenever .

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.

Bayesian society

It is maybe a more flexible society one which is guided by a variable ideology “I”,  fine-tuned continuously by bayesian techniques. The individual would be replaced by the bayesian individual, which forms its opinions from informations coming through a controlled channel. The input informations are made more or less available to the individual by using again bayesian analysis of interests, geographical location and digital footprint (creative commons attribution 2.0 licence, free online), closing the feedback loop.

A difference which makes four differences, in two ways

Gregory Bateson , speaking about the map-territory relation

“What is in the territory that gets onto the map? […] What gets onto the map, in fact, is difference.

A difference is a very peculiar and obscure concept. It is certainly not a thing or an event. This piece of paper is different from the wood of this lectern. There are many differences between them, […] but if we start to ask about the localization of those differences, we get into trouble. Obviously the difference between the paper and the wood is not in the paper; it is obviously not in the wood; it is obviously not in the space between them .

A difference, then, is an abstract matter.

Difference travels from the wood and paper into my retina. It then gets picked up and worked on by this fancy piece of computing machinery in my head.

… what we mean by information — the elementary unit of information — is a difference which makes a difference.

(from “Form, Substance and Difference”, Nineteenth Annual Korzybski Memorial
Lecture delivered by Bateson on January 9, 1970, under the auspices of the Institute of General Semantics, re-printed from the General Semantics Bulletin, no.
37, 1970, in Steps to an Ecology of Mind (1972))

This “difference which makes a difference” statement is quite famous, although sometimes considered only a figure of speach.

I think it is not, let me show you why!

For me a difference can be interpreted as an operator which relates images of the same thing (from the territory) viewed in two different maps, like in the following picture:

This figure is taken from “Computing with space…” , see section 1 “The map is the territory” for drawing conventions.

Forget now about maps and territories and concentrate on this diagram viewed as a decorated tangle. The rules of decorations are the following: arcs are decorated with “x,y,…”, points from a space, and the crossings are decorated with epsilons, elements of a commutative group (secretly we use an emergent algebra, or an uniform idempotent right quasigroup, to decorate arcs AND crossings of a tangle diagram).

What we see is a tangle which appears in the Reidemeister move 3 from knot theory. When epsilons are fixed, this diagram defines a function called (approximate) difference.

Is this a difference which makes a difference?

Yes, in two ways:

1. We could add to this diagram an elementary unknot passing under all arcs, thus obtaining the diagram

Now we see four differences in this equivalent tangle: the initial one is made by three others.
The fact that a difference is selfsimilar is equivalent with the associativity of the INVERSE of the approximate difference operation, called approximate sum.

2. Let us add an elementary unknot over the arcs of the tangle diagram, like in the following figure

called “difference inside a chora” (you have to read the paper to see why). According to the rules of tangle diagrams, adding unknots does not change the tangle topologically (although this is not quite true in the realm of emergent algebras, where the Reidemeister move 3 is an acceptable move only in the limit, when passing with the crossing decorations to “zero”).

By using only Reidemeister moves 1 and 2, we can turn this diagram into the celtic looking figure

which shows again four differences: the initial one in the center and three others around.

This time we got a statement saying that a difference is preserved under “infinitesimal parallel transport”.

So, indeed, a difference makes four differences, in at least two ways, for a mathematician.

If you want to understand more from this crazy post, read the paper 🙂

Rigidity of algebraic structure: principle of common cause

I follow with a lot of interest the stream of posts by Terence Tao on the Hilbert’s fifth problem and I am waiting impatiently to see how it connects with the field of approximate groups.

In his latest post Tao writes that

… Hilbert’s fifth problem is a manifestation of the “rigidity” of algebraic structure (in this case, group structure), which turns weak regularity (continuity) into strong regularity (smoothness).

This is something amazing and worthy of exploration!
I propose the following “explanation” of this phenomenon, taking the form of the:

Principle of common cause: an uniformly continuous algebraic structure has a smooth structure because both structures can be constructed from an underlying emergent algebra (introduced here).

Here are more explanations (adapted from the first paper on emergent algebras):

A differentiable algebra, is an algebra (set of operations A) over a manifold X with the property that all the operations of the algebra are differentiable with respect to the manifold structure of X. Let us denote by D the differential structure of the manifold X.
From a more computational viewpoint, we may think about the calculus which can be
done in a differentiable algebra as being generated by the elements of a toolbox with two compartments “A” and “D”:

– “A” contains the algebraic information, that is the operations of the algebra, as
well as algebraic relations (like for example ”the operation ∗ is associative”, or ”the operation ∗ is commutative”, and so on),
– “D” contains the differential structure informations, that is the information needed in order to formulate the statement ”the function f is differentiable”.
The compartments “A” and “D” are compatible, in the sense that any operation from “A” is differentiable according to “D”.

I propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq).

Algebraically, irqs are related with racks and quandles, which appear in knot theory (the axioms of a irq correspond to the first two Reidemeister moves). An uniform  irq is a family of irqs indexed by elements of a commutative group (with an absolute), such that  the third Reidemeister move is related to a statement in terms of uniform limits of composites of operations of the family of irqs.

An emergent algebra is an algebra A over the uniform irq X such that all operations and algebraic relations from A can be constructed or deduced from combinations of operations in the uniform irq, possibly by taking limits which are uniform with respect to a set of parameters. In this approach, the usual compatibility condition between algebraic information and differential information, expressed as the differentiability of algebraic operations with respect to the differential structure, is replaced by the “emergence” of algebraic operations and relations from the minimal structure of a uniform irq.

Thus, for example, algebraic operations and the differentiation operation (taking   the triple (x,y,f) to Df(x)y , where “x, y” are  points and “f” is a function) are expressed as uniform limits of composites of more elementary operations. The algebraic operations appear to be differentiable because of algebraic abstract nonsense (obtained by exploitation of the Reidemeister moves) and because of the uniformity assumptions which allow us to freely permute limits with respect to parameters in the commutative group (as they tend to the absolute), due to the uniformity assumptions.

The structure of visual space

Mark Changizi has an interesting post “The Visual Nerd in You Undestands Curved Space” where he explains that spherical geometry is relevant for the visual perception.

At some point he writes a paragraph which triggered my post:

Your visual field conforms to an elliptical geometry!

(The perception I am referring to is your perception of the projection, not your perception of the objective properties. That is, you will also perceive the ceiling to objectively, or distally, be a rectangle, each angle having 90 degrees. Your perception of the objective properties of the ceiling is Euclidean.)

Is it true that our visual perception senses the Euclidean space?

Look at this very interesting project [update 2021: link not working]

The structure of optical space under free viewing conditions

and especially at this paper:

The structure of visual spaces by J.J. Koenderink, A.J. van Doorn, Journal of mathematical imaging and vision, Volume: 31, Issue: 2-3 (2008), pp. 171-187

In particular, one of the very nice things this group is doing is to experimentally verify the perception of true facts in projective geometry (like this Pappus theorem).

From the abstract of the paper: (boldfaced by me)

The “visual space” of an optical observer situated at a single, fixed viewpoint is necessarily very ambiguous. Although the structure of the “visual field” (the lateral dimensions, i.e., the “image”) is well defined, the “depth” dimension has to be inferred from the image on the basis of “monocular depth cues” such as occlusion, shading, etc. Such cues are in no way “given”, but are guesses on the basis of prior knowledge about the generic structure of the world and the laws of optics. Thus such a guess is like a hallucination that is used to tentatively interpret image structures as depth cues. The guesses are successful if they lead to a coherent interpretation. Such “controlled hallucination” (in psychological terminology) is similar to the “analysis by synthesis” of computer vision.

So, the space is perceived to be euclidean based on prior knowledge, that is because prior controlled hallucinations led consistently to coherent interpretations.

Read arXiv every day? Yes!

This post from the Secret Blogging Seminar led me to this (now closed) question by Igor Pak at the Mathoverflow

Downsides of using the arxiv?

When my blood pressure went back to normal after reading the “downsides”, I spent some time informing myself about the answers given to this question, others than the very pertinent ones, in my opinion, from the Mathoverflow page. I think the best page to browse is the discussion from the meta.mathoverflow.net .

I have to say that really, yes!, I read arXiv every day, because it gives access to a lot of mathematical informations, which I filter according to my mathematical “nose” and not on an authority basis. One has to read papers in order to have an informed opinion.

The beautiful discussion page from meta.mathoverflow.net is an excellent example of the superiority of the new ways as compared with the older ones.

UPDATE(22.07.2011): The AMS Notices (August 2011) paper The changing Nature of Mathematical Publication is relevant for the subject of the post from the Secret Blogging Seminar. It appears to me that more or less the same strange problems concerning the arxiv are put forward in the article. Particularly this passage

If we ultimately publish our paper in a traditional journal, then how will that journal view our paper being first put on arXiv? If someone plagiarizes your work from arXiv, then what protections do you have?

seems to me to imply that there is less protection against plagiarism from arxiv than against plagiarism from traditionally published work. My take is that a paper on arxiv is more protected against plagiarism than a traditionally published paper, especially if you are not part of a politically strong team or country, because it is straightforward to prove the plagiarism (anyway easier than by relying on the publishing business and the peer review process). Besides, the “rhetorical question” seems to imply that it is not clear if there are any specific protections, like copyright, when in fact there are easy to find and clearly stated!

To finish, the purpose of the article is to announce a new publication column, “Scripta Manent”. The peer review process for this paper failed to notice that the title of this new publication column is spelled “Scripta Manet” twice!

Maps in the brain: fact and explanations

From wikipedia

Retinotopy describes the spatial organization of the neuronal responses to visual stimuli. In many locations within the brain, adjacent neurons have receptive fields that include slightly different, but overlapping portions of the visual field. The position of the center of these receptive fields forms an orderly sampling mosaic that covers a portion of the visual field. Because of this orderly arrangement, which emerges from the spatial specificity of connections between neurons in different parts of the visual system, cells in each structure can be seen as forming a map of the visual field (also called a retinotopic map, or a visuotopic map).

See also tonotopy for sounds and the auditory system.

The existence of retinotopic maps is a fact, the problem is to explain how they appear and how they function without falling into the homunculus fallacy, see my previous post.

One of the explanations of the appearance of these maps is given by Teuvo Kohonen.

Browse this paper (for precise statements) The Self-Organizing map , or get a blurry impression from this wiki page. The last paragraph from section B. Brain Maps reads:

It thus seems as if the internal representations of information in the brain are generally organized spatially.

Here are some quotes from the same section, which should rise the attention of a mathematician to the sky:

Especially in higher animals, the various cortices in the cell mass seem to contain many kinds of “map” […] The field of vision is mapped “quasiconformally” onto the primary visual cortex. […] in the visual areas, there are line orientation and color maps. […] in the auditory cortex there are the so-called tonotopic maps, which represent pitches of tones in terms of the cortical distance […] at the higher levels the maps are usually unordered, or at most the order is a kind of ultrametric topological order that is not easy interpretable.

Typical for self-organizing maps is that they use (see wiki page) “a neighborhood function to preserve the topological properties of the input space”.

From the connectionist viewpoint, this neighbourhood function is implemented by lateral connections between neurons.

For more details see for example Maps in the Brain: What Can We Learn from Them? by Dmitri B. Chklovskii and Alexei A. Koulakov. Annual Review of Neuroscience 27: 369-392 (2004).

Also browse Sperry versus Hebb: Topographic mapping in Isl2/EphA3 mutant mice by Dmitri Tsigankov and Alexei A. Koulakov .

Two comments:

1. The use of a neighbourhood function is much more than just preserving topological information. I tentatively propose that such neighbourhood functions appear out of the need of organizing spatial information, like explained in the pedagogical paper from the post Maps of metric spaces.

2. Just to reason on discretizations (like hexagonal or other) of the plane is plain wrong, but this is a problem encountered in many many places elsewhere. It is wrong because it introduces the (euclidean) space on the back door (well, this and using happily an L^2 space).

Maps of metric spaces

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its “physical” meaning, and dilation structures as a convenient simplification of an exhaustive database of maps of a metric space into another:

Maps of metric spaces

The material is taken and slightly adapted from the long paper “Computing with space”, check for updates of this on this page.

Smooth geometry vs (nonsmooth calculus and combinatorics)

I am intrigued by this part of the  post from NEW

“The public talk by Cumrun Vafa puts out the classic message that strings have come to the rescue of physics, unifying QM and gravity, and that:

Smooth geometry of strings seems to explain all known interactions (at least in principle)”

(my emphasis)

Why “smooth”? Probably only because this is in the comfort zone of many.

However, there are two new fields of mathematics which deserve to be taken into consideration by physicists (or not, not my problem in fact):

• Nonsmooth calculus, see for an intro this excellent review by  Juha Heinonen .
• Additive combinatorics,  see this book by Terence Tao, Van Vu.

  That’s the future!

UPDATE:  (24.03.2012) Congratulations to Endre Szemeredi, the Abel Prize Laureate 2012, “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.”

… and modern physics, maybe in 50 years.

arXiv for Cezanne

In the middle of the 19th century, in France, just before the impressionist revolution,  painting was boring. Under the standards imposed by the Academie des Beaux Arts, paintings were done in a uniform technique, concerning a very restrictive list of subjects. I cite from wikipedia:

”Colour was somber and conservative, and the traces of brush strokes were suppressed, concealing the artist’s personality, emotions, and working techniques.”

This seems very much similar with the situation in today’s mathematical research.

Read the rest here: “Boring mathematics, artistes pompiers and impressionists”.

PS: anyone who understands from this post that I think mathematics is boring has the attention span of a gnat.

Hilbert fifth’s problem without one parameter subgroups

Further I reproduce, with small modifications, a comment   to the post

Locally compact groups with faithful finite-dimensional representations

by Terence Tao.

My motivation lies in the  project   described first time in public here.  In fact, one of the reasons to start this blog is to have a place where I can leisurely explain stuff.

Background:    The answer to the  Hilbert fifth’s problem  is: a connected locally compact group without small subgroups is a Lie group.

The key idea of the proof is to study the space of one parameter subgroups of the topological group. This space turns out to be a good model of the tangent space at the neutral element of the group (eventually) and the effort goes towards turning upside-down this fact, namely to prove that this space is a locally compact topological vector space and the “exponential map”  gives a chart  of  (a neighbourhood of the neutral element of ) the group into this space.

Because I am a fan of differential structures   (well, I think they are only the commutative, boring side of dilation structures  or here or emergent algebras)   I know a situation when one can prove the fact that a topological group is a Lie group without using the one parameter subgroups!

Here starts the original comment, slightly modified:

Contractive automorphisms may be as relevant as one-parameter subgroups for building a Lie group structure (or even more), as shown by the following result from E. Siebert, Contractive Automorphisms on Locally Compact Groups, Math. Z. 191, 73-90 (1986)

5.4. Proposition. For a locally compact group G the following assertions are equivalent:
(i) G admits a contractive automorphism group;
(ii) G is a simply connected Lie group whose Lie algebra g admits a positive graduation.

The corresponding result for local groups is proved in L. van den Dries, I. Goldbring, Locally Compact Contractive Local Groups, arXiv:0909.4565v2.

I used Siebert result for proving the Lie algebraic structure of the metric tangent space to a sub-riemannian manifold in M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136. arXiv:0804.0135v2

(added here: see  in Corollary 6.3 from “Infinitesimal affine …” paper, as well as Proposition 5.9 and Remark 5.10 from 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 , arXiv:0810.5042v4 )

When saying that contractive automorphisms, or approximately contractive automorphisms [i.e. dilation structures], may be more relevant than one-parameter subgroups, I am thinking about sub-riemannian geometry again, where a one-parameter subgroup of a group, endowed with a left-invariant distribution and a corresponding Carnot-Caratheodory distance, is “smooth” (with respect to Pansu-type derivative) if and only if the generator is in the distribution. Metrically speaking, if the generator  is not in the distribution then any trajectory of the one-parameter group has Hausdorff dimension greater than one. That means lots of problems with the definition of the exponential and any reasoning based on differentiating flows.

The Cartesian Theater: philosophy of mind versus aerography

UPDATE (2021): Now collected in Gnomons and homunculi in the space theater, among other writings.

Looks to me there is something wrong with the Cartesian Theater term.

Short presentation of the Cartesian Theater, according to wikipedia (see previous link):

The Cartesian theater is a derisive term coined by philosopher Daniel Dennett to pointedly refer to a defining aspect of what he calls Cartesian materialism, which he considers to be the often unacknowledged remnants of Cartesian dualism in modern materialistic theories of the mind.

Descartes originally claimed that consciousness requires an immaterial soul, which interacts with the body via the pineal gland of the brain. Dennett says that, when the dualism is removed, what remains of Descartes’ original model amounts to imagining a tiny theater in the brain where a homunculus (small person), now physical, performs the task of observing all the sensory data projected on a screen at a particular instant, making the decisions and sending out commands.

Needless to say, any theory of mind which can be reduced to the Cartesian Theater is wrong because it leads to the homunculus fallacy: the homunculus has a smaller homunculus inside which is observing the sensory data, which has a smaller homunculus inside which …

This homunculus problem is very important in vision. More about this in a later post.

According to Dennett, the problem with the Cartesian theater point of view is that it introduces an artificial boundary (from Consciousness Explained (1991), p. 107)

“…there is a crucial finish line or boundary somewhere in the brain, marking a place where the order of arrival equals the order of “presentation” in experience because what happens there is what you are conscious of.”

As far as I understand, this boundary creates a duality: on one side is the homunculus, on the other side is the stage where the sensory data are presented. In particular this boundary acts as a distinction, like in the calculus of indications of Spencer-Brown’ Laws of Form.

This distinction creates the homunculus, hence the homunculus fallacy. Neat!

Why I think there is something wrong with this line of thought? Because of the “theater” term. Let me explain.

The following is based on the article of Kenneth R Olwig

“All that is landscape is melted into air: the `aerography’ of ethereal space”, Environment and Planning D: Society and Space 2011, volume 29, pages 519 – 532.

but keep in mind that what is written further represents my interpretation of some parts of the article, according to my understanding, and not the author point of view.

There has been a revolution in theater, started by

“…the early-17th-century court masques (a predecessor of opera) produced by the author Ben Jonson (the leading author of the day after Shakespeare) together with the pioneering scenographer and architect Inigo Jones.
The first of these masques, the 1605 Masque of Blackness (henceforth Blackness ), has a preface by Jonson containing an early use of landscape to mean scenery and a very early identification of landscape with nature (Olwig, 2002, page 80), and Jones’s scenography is thought to represent the first theatrical use of linear perspective in Britain (Kernodle, 1944, page 212; Orgel, 1975).” (p. 521)Ben Johnson,

So? Look!

“From the time of the ancient Greeks, theater had largely taken place outside in plazas and market places, where people could circle around, or, as with the ancient Greco-Roman theater or Shakespeare’s Globe, in an open roofed arena. Jones’s masques, by contrast, were largely performed inside a fully enclosed rectangular space, giving him control over both the linear-focused geometrical perspectival organization of the performance space and the aerial perspective engendered by the lighting (Gurr, 1992; Orrell, 1985).” (p. 522, my emphasis)

“Jonson’s landscape image is both enframed by, and expressive of, the force of the lines of perspective that shoot forth from “the eye” – notably the eye of the head of state who was positioned centrally for the best perspectival gaze.” (p. 523, my emphasis)

“Whereas theater from the time of the ancient Greeks to Shakespeare’s Globe was performed in settings where the actor’s shadow could be cast by the light of the sun, Jones’s theater created an interiorized landscape in which the use of light and the structuring of space created an illusion of three dimensional space that shot from the black hole of the individual’s pupil penetrating through to a point ending ultimately in ethereal cosmic infinity. It was this space that, as has been seen, and to use Eddington’s words, has the effect of “something like a turning inside out of our familiar picture of the world” (Eddington, 1935, page 40). It was this form of theater that went on to become the traditional `theater in a box’ viewed as a separate imagined world through a proscenium arch.” (p. 526, my emphasis)

I am coming to the last part of my argument: Dennett’ Cartesian Theater is a “theater in a box”. In this type of theater there is a boundary,

“… scenic space separated by a limen (or threshold) from the space of the spectators – today’s `traditional’ performance space [on liminality see Turner (1974)]” (p. 522)

a distinction, as in Dennett argument. We may also identify the homunculus side of the distinction with the head of state.

But this is not all.

Compared with the ancient Greeks theater, the “theater in a box” takes into account the role of the spectator as the one which perceives what is played on stage.

Secondly, the scenic space is not “what happens there”, as Dennett writes, but a construction already, a controlled space, a map of the territory and not the territory itself.

Conclusion: in my view (contradict me please!) the existence of the distinction (limen) in the “Cartesian theater”, which creates the homunculus problem, is superficial. More important is the fact that “Cartesian theater”, as “theater in a box”, is already a representation of perception, having on one side of the limen a homunculus and on the other side a scenic space which is not the “real space” (as for example the collection of electric sparks sent by the sensory organs to the brain) but instead is as artificial as the homunculus, being a space created and controlled by the scenographer.

Litmus test: repeat the reasoning of Dennett after replacing the “theater in a box” preconception of the “theater” by the older theater from the time of ancient Greeks. Can you do it?

On the beautiful idea of “aerography”, later.

Book project: Metric spaces with dilations

On the mathematics front, here is a link to a book project

Metric spaces with dilations

(working version! chech for updates)

As it is now, it’s not much more than a merger of previous research papers, without the nice figures from “Computing with space…” , but from seeing all in one place you may get a sense of what is this all about.

This first version has been prepared as a basis for the minicourse

“Carnot-Caratheodory spaces as metric spaces with dilations”

held at a January 2011 Summer School in Rio de Janeiro.

In the same place I started being interested in exploring this frontier between neuroscience and mathematics…