Non-Euclidean analysis, a bit of recent history

Being an admirer of bold geometers who discovered that there is more to geometry than euclidean geometry, I believe that the same is true for analysis. In my first published paper “The topological substratum of the derivative” (here is a scan of this hard to find paper), back in 1993, I advanced the idea that there are as many “analyses” as the possible fields of dilations, but I was not aware about Pierre Pansu huge paper from 1989 “Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un” (sorry for the missing accents, I am still puzzled by the keyboard of the computer I am using to write this post), where he invents what is now called “Pansu calculus”, which is the analysis associated to a Carnot group.

The same idea is then explored in the papers “Sub-riemannian geometry and Lie groups, I“, “Tangent bundles to sub-riemannian groups“, “Sub-riemannian geometry and Lie groups II“. These papers have not been published (only put on arXiv), because at that moment I hoped that the www will change publishing quick (I still do believe this, but now I am just a bit wiser, or forced by bureaucracy to publish or perish), so one could communicate not only the very myopic technical, incremental result, but also the ideas behind, the powerful  meme.

During those years (2001-2005) I have been in Lausanne, trying to propagate the meme around, in Europe, as I said previously. There were mixed results, people were not taking this serious enough, according to my taste. Sergey Vodopyanov had ideas which were close to mine, except that he was trying to rely on what I call “euclidean analysis”, instead of intrinsic techniques, as witnessed by his outstanding results concerning detailed proofs in low-regularity sub-riemannian geometry. (I was against such results by principle, because what is C^{1,1} but euclidean regularity? but the underlying ideas were very close indeed).

In a very naive way I tried to propagate the meme further, by asking for a visit at IHES, in 2004, when I had the pleasure to meet Pierre Pansu and Andre Bellaiche, then I dared to ask for another visit immediately and submitted the project

“Non-Euclidean Analysis” start-up

which I invite you to read. (The project was rejected, for good reasons, I was already there visiting and suddenly I was asking for another, much longer visit)

Then, from 2006 I went back to basics and proposed axioms for this, that is how dilation structures appeared (even if the name and a definition containing the most difficult axiom was already proposed in the previous series of papers on sub-riemannian geometry and Lie groups.  See my homepage for further details and papers (published this time).

I see now that, at least at the level of names of grant projects, the meme is starting to spread. Here is the “Sub-riemannian geometric analysis in Lie groups” GALA project and here is the more recent “Geometric measure theory in Non Euclidean spaces” GeMeThNES project.

Bipotentials, variational formulations

The paper on the use of bipotentials in variational formulations is finally submitted, also available on arxiv here. See also this presentation.

In case you wonder how this could be related with other subjects commented on this blog, then wait to see “A gallery of emergent algebras”, where it shall be explained the connection between convex analysis and an emergent algebra related to a semidirect product between the semigroup of words over a symplectic space and \mathbb{R}.