If X is homogeneous, analytic, and strongly locally homogeneous, then there is an analytic group acting transitively on X. There is an example of an analytic space on which some separable metrizable group acts transitively, but on which no analytic group acts transitively.

I aim to give the flavour of a selection of topics based around Fräıssé’s notion of homogeneous structure. This is an area connecting ideas from model theory, permutation group theory, combinatorics, descriptive set theory, complexity theory, and other subjects. I will touch superficially a number of these subjects, with little in depth. The background assumed will be basic first order logic (l...

In this paper we extend the De Giorgi notion of rectifiability of surfaces in non homogeneous Lie groups. This notion and the principal properties of Cacciopoli sets had already been proved in homogeneous Lie group, using a blow-up method, with respect to the natural dilations. In non homogeneous Lie groups no dilation are defined, so that we need to apply a freezing method, locally approximati...