Proof mining and approximate groups, announcement
The last weeks have been very busy for personal reasons. I shall come back to writing on this blog in short time.
With Laurentiu Leustean we won (a month ago, but the project has been submitted this Spring) the financing for our research project
“Proof mining in metric anaysis, geometric group theory and ergodic theory”
(project PN-II-ID-PCE-2011-3-0383). Laurentiu is a specialist in proof mining with applications to geodesic spaces and ergodic theory; I am interested in emergent algebras, particularly in dilation structures, so one of our aims (in this project) is to understand why nilpotent like structures appear in the famous Gromov theorem on groups of polynomial growth, as well in approximate groups, by using proof mining techniques for “finitizing” emergent algebras, roughly.
This program is very close to one of the programs of Terence Tao, who continues his outstanding research on approximate groups. The following post
made me happy because it looks like confirming that our dreams (for the moment) have a correspondent in reality and probably ideas like this are floating in the air.
UPDATE 25.10: Today a new post of Tao announces the submission on arxiv of the paper by him, Emmanuel Breuillard and Ben Green, “The structure of approximate groups“. I look forward to study it, to see if they explain why nilpotent structures appear in the limit. My explanation, in a different context, “A characterization of sub-riemannian spaces…”, related also to work of Gromov, namely why in sub-riemannian geometry nilpotent groups appear as models of metric tangent spaces, is that this is a feature of an emergent algebra. See also previous posts, like Principles: randomness/structure or emergent from a common cause?.