According to the following Journal Articles, there are these structures called Dilation Structures that are formalised in Emergent Algebras, examined in the case of metric spaces with dilations, and are basically structures that are partly group-like and partly manifold-like.

I can't find in these articles or other related articles any applications of these Dilation Structures and Emergent Algebras on the whole. Know where I might be able to read up on the applications?

The author here. Emergent algebras and dilation structures appeared from the effort to find an answer to a problem by Gromov in sub-riemannian (aka Carnot-Caratheodory) geometry: "Develop a sufficiently rich and robust internal CC language which would enable us to capture the essential external characteristics of our CC spaces". Here is a short description with links of the problem.

This problem is to recover the differential structure from the metric structure, which is a matter of analysis more than one of geometry. In the particular case of riemannian geometry this was solved by Nikolaev in 1998-1999.

A solution to this problem is in this article, which uses dilation structures.

Another application is more algebraic, namely to characterize what could be the correspondent of affine geometry for Carnot groups, a particular class of nilpotent groups which generalize vector spaces (and they appear naturally in SR geometry, but also elsewhere, for example the Heisenberg groups are the simplest non-commutative Carnot groups).

Yet another application, which is work in progress, is to treat the formalism of emergent algebras as yet another interaction graphs theory.

Marius Buliga has considered applications of dilation structures in the context of image processing by the visual system, see What is a space? Computations in emergent algebras and the front end visual system (2010), and More than discrete or continuous: a bird's view.

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