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

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):

  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.

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