On forking and definability of types in some dp-minimal theories

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.Comment: Appeared previously as an appendix in arXiv:1210.447

Some remarks on locally o-minimal structures (Model theoretic aspects of the notion of independence and dimension)

Locally o-minimal structures are some local adaptation from o-minimal structures. They were treated, e.g. in [1], [2]. We try to characterize types of definably complete locally o-minimal structures. And we argue about the dp-rank of them

Connected components of definable groups and o-minimality I

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact first order theories.Comment: 26 pages. This paper corrects the paper "Groups definable in o-minimal structures: structure theorem, G^000, definable amenability, and bounded orbits" by the first author which was posted in December (1012.4540v1) and later withdraw

A note on generic subsets of definable groups

We generalize the theory of generic subsets of definably compact de-finable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.Fundação para a Ciência e a Tecnologia, Financiamento Base 2008 - ISFL/1/20

Spectral Spaces in o-minimal and other NIP theories

We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction