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

July 21, 2012 chorasimilarity 2 Comments

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.

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2004: Fleeced, 2012: The cost of knowledge

July 18, 2012 chorasimilarity Leave a comment

In 2004 Rob Kirby publishes in the Notices of the AMS the opinion article “Fleeced?“. Let me quote a bit from it.

“Most mathematicians feel that they own their journals. They write and submit papers to their favorite (often specialized) journals. They often referee for those same journals. And some devote time and energy as editors. Throughout this process there is no contact with nonmathematicians, except for some of the editors. It is no wonder that mathematicians have a sense of pride and ownership in their journals.

But the truth is that, legally, mathematicians do not own the commercial journals. Elsevier and Academic Press journals are a highly profitable part of a big corporation. Bertelsmann has recently divested Springer, and now Springer, Kluwer, and Birkhäuser are owned by an investment company (who did not buy these publishers in order to make less profit than before). […]

We mathematicians simply give away our work (together with copyright) to commercial journals who turn around and sell it back to our institutions at a magnificent profit. Why? Apparently because we think of them as our journals and enjoy the prestige and honor of publishing, refereeing, and editing for them. […]

What can mathematicians do? At one extreme they can refuse to submit papers, referee, and edit for the high-priced commercial journals. At the other extreme they can do nothing. […]

A possibility is this: one could post one’s papers (including the final version) at the arXiv and other websites and refuse to give away the copyright. If almost all of us did this, then no one would have to subscribe to the journals, and yet they could still exist in electronic form.
Personally, I (and numerous others) will not deal with the high-priced journals. What about you?”

In 2012 appeared The cost of knowledge, a site inspired by the blog post Elsevier – my part in its downfall by Timothy Gowers.

In 12 years the world changed a bit in this respect. It will change much more.

Let me finish this post by describing my modest experience related to this subject, during these 12 years.

In 2004 I was a kind of a post-doc/visitor (on a contract which was prolonged once a year, for a max of 6 years) at EPFL (Lausanne, Swiss). I already decided some years ago to act as if the future of mathematical publication is the arxiv and alike. One reason is the obvious fact that the www will change the world much more than the invention of the press did. Almost all research which was left in manuscript perished after the press revolution. Everybody who wants to give something to the research community has to put its research on the net, I thought, and simultaneously, to help the old system to die, by not publishing in paper journals. Moreover, I had troubles in the past with publishing multidisciplinary papers. I always believed that it is fun to mix in a paper several fields, that there is one mathematics, and so on, but such papers were extremely difficult to publish, at least these papers written by me, with my modest competence (and Romanian origin, I have to say this). So, lulled by the relative swiss security, I was just putting my papers on arxiv, moreover written in an open form which was inviting others to participate to the same research, at least that I was thinking.

The result? In 2004-2005 I was practically laughed into the face. Who cares about a paper in arxiv, which is not published in journal form?

In 2006 I returned in Romania, decided to start to publish in paper journals, because what I was trying to do was either disregarded as not counting, or discretely and “creatively” borrowed. I could not renounce to my believes, therefore I arrived to a system of waves of papers in arxiv, some of them sent to publication (so to say, the most conservative ones).

After a time I started to recover after my strategic “fault”, but still there is work to do. But is it right to be forced to hide own beliefs? Apparently, I am right in my beliefs, which are similar to those publicly declared by great mathematicians, at least since 2004. Practically, a big chunk of my career was/is still disturbed by this immense inertia. I am surely just an example among many others colleagues who are suffering similar experiences.

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Approximate groupoids may be useful

July 11, 2012 chorasimilarity 1 Comment

UPDATE (Jan 4 2013): See also this: Approximate groupoids again.

____________________________________

I appreciated yesterday the talk of Harald Helfgott, here is the link to the arxiv paper which was the subject of his communication: On the diameter of permutation groups (with Á. Seress).

At the end of his talk Harald stated as a conclusion that (if I well understood) the approximate groups field should be regarded as the study of

“stable configurations under the action of a group”

where “stable configuration” means a set (or a family…) with some controlled behaviour of its growth under the “action of a group”, understood maybe in a lax, approximate sense.

This applies to approximate groups, he said, where the action is by left translations, but it could apply to the conjugate action as well, and evenin more general settings, where one has a space where a group acts.

My understanding of this bold claim is that Harald suggests that the approximate groups theory is a kind of geometry in the sense of Felix Klein: the (approximate, in this case) study of stable configuratiuons under the action of a group.

Very glad about hearing this!

Today I just remembered a comment that I have made last november on the blog of Tao, where I proposed to study “approximate groupoids”.

Why is this related?

Because a group acting on a set is just a particular type of groupoid, an action groupoid.

Here is the comment (link), reproduced further.

“Question (please erase if not appropriate): as a metric space is just a particular example of a normed groupoid, could you point me to papers where “approximate groupoids” are studied? For starters, something like an extension of your paper “Product set estimates for non-commutative groups”, along the lines in the introduction “one also consider partial sum sets …”, could be relevant, but I am unable to locate any. Following your proofs could be straightforward, but lengthy. Has this been done?

For example, a $k$ approximate groupoid $A \subset G$ of a groupoid $G$ (where $G$ denotes the set of arrows) could be a

(i)- symmetric subset of $G$ : $A = A^{-1}$ , for any $x \in Ob(G)$ $id_{x} \in A$ and

(ii)- there is another, symmetric subset $K \subset G$ such that for any $(u,v) \in (A \times A) \cap G^{(2)}$ there are $(w,g) \in (A \times K) \cap G^{(2)}$ such that $uv = wg$ ,

(iii)- for any $x \in Ob(G)$ we have $\mid K_{x} \mid \leq k$ .

One may replace the cardinal, as you did, by a measure (or random walk kernel, etc), or even by a norm.”

UPDATE (28.10.2012): Apparently unaware about the classical proof of Ronald Brown, by way of groupoids, of the Van Kampen theorem on the fundamental group of a union of spaces, Terence Tao has a post about this subject. I wonder if he is after some applications of his results on approximate groups in the realm of groupoids.

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On plagiarism scandals in Romania, an easy test

July 3, 2012 chorasimilarity 2 Comments

What the naked emperors from Romanian politics and universities don’t understand is that, today, it is a very easy way to check the credibility of statements: show me a link to your arguments and let me decide. That is all.

It goes totally against (hollow) authority arguments, therefore it takes away power (to do harm), and that pisses a lot of people, worried about the fact that everybody might see their incompetence and stupidity.

Here are some links:

Nature: Romanian prime minister accused of plagiarism
Nature: Romanian panel calls prime minister a plagiarist . But committee is dissolved during the course of its meeting.

Link to pdf with plagiarized content (after searching 1 min on the net)

According to politicians and, amazingly, according to some “scientists”, there is a political motivation behind the articles from Nature.

Disgusting.

Finally, there is a simple test, which I suggest you to apply to any scientist: don’t believe the titles, prizes and so on. Instead, look after the articles. Does the guy have his/her articles freely available (on the homepage, on arxiv, etc)? If not, then DANGER!

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Monthly Archives: July 2012

2004: Fleeced, 2012: The cost of knowledge

On plagiarism scandals in Romania, an easy test

computing with space | open notebook