I don’t know how you function, but I use to send messages, containing ideas which turn around in my head, to whom I think might be interested, as an invitation to dialogue, or as a request for help.
Reading such messages afterwards, almost always I notice that it is not obvious what are my intentions and motivations, even to me.
Nevertheless, on a longer time scale, these become transparent. Some look a bit like messages in a bottle which I send from a remote island and years after I receive them on another shore.
These poetic lines, partially motivated by the sun intake and the proximity of the sea, are an introduction to such a bottled message, which I just received and moreover one which can be used as evidence for the long time scale phenomenon.
I use now an older laptop, the one which I had with me when I was in Rio. I don’t know if I wrote this before, or if anybody cares (but indulge me to use this place as a log, besides being an open notebook), the strongest feeling I had back in Rio was one of total freedom. And what does a free mathematician these days? Well, I decided I would like to switch to biology, of course.
I have a folder on my laptop, created in Rio, called “next”. What’s that? I asked myself today and opened it. Inside there’s a latex file “bio_next”. Aha, let’s see, I don’t remember what ‘s about.
In the file is a collection of quotes from messages sent to somebody I shall not mention, because they are part of strange messages which probably did not have any meaning from the receiver point of view.
But I invite you to compare these messages with the content of the chorasimilarity open notebook, in order to prove that there’s a long time scale coherence. Hopefully, now I am a bit wiser and I keep a log.
I shall mention also, because as somebody said that the net is not subtle, that the purpose of sharing these messages is to invite you, dear reader, to dialogue.
Here is the content of the file (latex commands and names/places excluded):
nov 14 2007: I hope to be in .. at the beginning of December (but not 100 percents sure). I could pass by … if you are there, because I am very curious about your opinion. Mine is that even these contributions are unworthy, they show the existence of an infinite field of analysis. So maybe this could nuance the old dichotomy discrete-continuum a bit.
feb 1, 2008: On another note, I am (was) impressed by your turning to biology and now am part of a project in molecular dynamics (more of less). So now I am strugling with questions like: what are “shape” and “function” meaning for a biochemist? Very non bourbakian notions.
oct 5, 2009: I think that the paper is difficult for someone not in the right frame of mind and rigid. I also believe that it is important, less because it gives a kind of axiomatization of SR geometry, but more because it shows that there is a world of analysis outside the regular paths. And I think is relevant for understanding our presupositions concerning the space, but I noticed that this is not a viewpoint to discuss in a math paper for publication.
nov 16, 2009: Question (to your curiosity): if a squirrel (monocular vision times two eyes) would do analysis, what it would look like? My tentative answer: the axiom A4 (with the approximate difference operator, the one from the last section of Bellaiche SR paper) of dilatation structures, seems natural to us, binocular vision beings. To a squirrel, the most natural is to base everything on the approximate inverse operator ( inv , see the previous “Emergent algebras as generalizations of differentiable algebras, with applications” http://arxiv.org/abs/0907.1520 ) Therefore, a squirrel would start to do analysis on symmetric spaces, not on vector spaces as us.
jan 5, 2010: Question: The basic component of dilatation structures (and therefore of any scheme, let’s say, of finite differences) are dilatations. A dilatation is just a gate with two entries and one output. The calculus with dilatations consists in finding interesting binary trees with nodes beind dilatations, such that when goes to 0, the output of the tree converges. Likewise, boolean calculus is based on the transistor (say, a NAND gate) and any circuit made by transistors is (in multiple ways) representable as a binary tree. The trees appearing in the axioms of dilatation structures are different from the ones appearing in the axioms of boolean algebras, but the question remains: find interesting circuits and their algebra, but for dilatation structures. So I have a question to ask you: if you think about dilatation gates as elementary molecules, what is their chemistry? That is, have you seen (binary) trees and/or algebraic rules involving such objects in chemistry? Is maybe process algebra another ingredient to add? For process algebra see http://www.lfcs.inf.ed.ac.uk/reports/89/ECS-LFCS-89-86/ and the wiki entry http://en.wikipedia.org/wiki/Process_calculus I would appreciate any help/idea.
feb 12, 2010: Otherwise, concerning my last mail, now I have an answer, namely dilations in metric spaces are approximate quandle operations (Reidemeister move III is true in the limit of the rescaling which gives the tangent space)! It is somehow natural, if you think about the Alexander quandle, which is an “exact” quandle due to the linearity, and the quandle operation is just an Euclidean dilation. Sorry that I can’t explain more precisely in few words, but this means that there is a whole algebraic avenue to explore!
jun 10, 2010: This is just to communicate some ideas about what means to simulate (computations in) a space, which I discovered that seems to be related to some viewpoints in studies of human vision, like Koenderink’ “Brain a geometry engine”.