Short Abstract:
Let X be a definable subset of some o-minimal structure. We study the spectrum of X, in relation wih the definability of types.

Long Abstract:
Let M be an o-minimal structure, and X be a definable subset of Mn. The spectrum of X is the set of complete n-types on X, with the topology whose basis is given by the open definable subsets. The spectrum was introduced by Pillay, as a generalisation of the real spectrum of semi-algebraic sets. If M expands a group, the spectrum is a normal spectral space. In particular, it is non-Hausdorff, and it is endowed with the specialisation order: p ≼ q (p is a specialisation of q or q is a generalisation of p) if p is in the closure of q.
(X, ≼) is a partial order, and for every type p, the set of its specialisations is totally ordered by ≼. We study the problem whether a type is rational or not, for M expanding a real closed field. We show that the answer is determined by the closure of a suitable normalisation q of p (with q and p mutually definable). More precisely, let r be the specialisation of q of smallest dimension. Then, q is rational iff r is realised. Moreover, q is rational over M(r), and every type realised in M(r) is either already realised in M, or it is irrational.

Abstract:
Modifying Fraissé's amalgamation method, Ehud Hrushovski constructed a strongly minimal set with a non-locally modular geometry which does not come from an algebraically closed field. He thus refuted the Zilber trichotomy. Using the same technique, he managed to "fuse" two strongly minimal sets into a single one. Hrushovski's amalgamation technique has been extended to constructions over vector spaces. This lead to the construction of a new uncountably categorical group (nilpotent of class 2) and more recently to the construction of various algebraically closed fields with extra structure, in particular a field of finite Morley rank with a distinguished proper infinite additive subgroup (in positive characteristic) and a field of finite Morley rank with a distinguished proper infinite multiplicative subgroup (in characteristic 0), i.e. a bad field. The latter is joint work with A. Baudisch, A. Martin-Pizarro and F. Wagner. We give a survey of the above results.

Abstract:
Cotorsion modules represent a homologically defined generalization of pure-injective modules. There are cotorsion abelian groups that are not pure-injective, but over von Neumann regular rings (that is, rings over which all modules have quantifier elimination) and over pure semisimple rings (that is, rings over which every module is totally transcendental), this generalization yields nothing new. Joint work with Ivo Herzog will be presented that characterizes the class of rings (containing the former two classes) over which every cotorsion module is pure-injective.

Abstract:
A Definably Complete Structure is an expansion of an ordered field, such that every definable subset of the domain which is bounded from above has a supremum. Every expansion of the real field has this property, by Dedekind completeness. We study the basic properties of the class of definably complete structures in a given language. This class, which is recursively axiomatized, expands the class of real closed fields and includes o-minimal expansions of ordered fields. This inclusion is however proper, and we can not, in general, use the powerful tools of o-minimality to prove our results. Our main result is a Theorem of Decomposition of Noetherian Varieties into finitely many smooth components. The generality of the setting and of the techniques used allows us to exhibit an example of a class of smooth-but-not-analytic real varieties for which the theorem holds.