Gareth O. Jones’s research while affiliated with University of Manchester and other places

... Proof. As G is definably connected and semisimple, there is an M-definable real closed field R such that G is definably isomorphic in M to a semialgebraic group over the field of real algebraic numbers R alg ⊆ R, by [HPP11, 4.4(ii)] or [EJP07]. In case (a) our claim follows from [HPP11, 6.3] and thus we only have to consider case (b). ...

Reference:

Cartan subgroups of groups definable in o-minimal structures

... In order to prove the torsion points result under weaker assumptions it seems important to develop similar topological tools to the ones originally used, but this time without the triangulation theorem. Indeed, Sheaf Cohomology in expansions of ordered groups has been the subject of several papers of Edmundo, Jones and Peatfield (see [13] and [14]) and of Beraducci and Fornasiero (see [4]). In [16], Edmundo and Terzo prove Pillay's conjecture under a relatively weak assumption on M but with an additional assumption of "orientability" on the group G. ...

Reference:

PILLAY’S CONJECTURE AND ITS SOLUTION-A SURVEY

... As shown in [11,Proposition 3.2], if A is a commutative ring, then the category Mod(A X def ) of sheaves of A-modules on X (relative to the o-minimal site) is isomorphic to the category Mod(A X ) of sheaves of A-modules on a certain spectral topological space X, the o-minimal spectrum of X, associated to X. The o-minimal spectrum X of a definable set X is the set of ultra-filters of definable subsets of X (also know in model theory as types concentrated on X) equipped with the topology generated by the subsets U with U ∈ Op(X def ). ...

Reference:

On Proper Direct Image in o-Minimal Expansions of Groups

... Nonetheless, in this paper we only consider natural interpretability without parameters. In fact, when M is an o-minimal expansion of a real closed field and G and Z are definable without parameters, by [16,Cor. 1.2], the notions of naturally interpretable with and without parameters agree. ...

Reference:

Finite central extensions of o-minimal groups