DEFINABLE AND INVARIANT TYPES IN ENRICHMENTS OF NIP THEORIES

Definable and Invariant Types in Enrichments of NIP Theories

Lascar strong types and forking in NIP theories

This is an updated and slightly expanded version of a tutorial given in the Mini-Course in Model Theory, Torino, February 9-11, 2011. Some parts were previously exposed in the Model Theory Seminar of Barcelona. The main goals were to clarify the relation of forking with some versions of splitting in NIP theories and to present the known results on G-compactness, including a full proof of a theo...

Type decomposition in NIP theories

A first order theory is NIP if all definable families of subsets have finite VCdimension. We provide a justification for the intuition that NIP structures should be a combination of stable and order-like components. More precisely, we prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and an order-like quotient.

On NIP and invariant measures

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd...

Definable Structures in O-minimal Theories: One Dimensional Types

Let N be a structure definable in an o-minimal structureM and p ∈ SN (N), a complete N -1-type. If dimM(p) = 1 then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N -definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to...