# Model Theory and Applications

## Scope of the special session

The topic of this special session is model theory, a branch of mathematical logic dealing with mathematical structures (models) from the point of view of first order logical definability. The original goal in model theory is to build, study and classify mathematical universes in which some given axioms (usually expressed in a first order way) are satisfied.

Model theory has remote roots in the birth of non-Euclidean geometries and in the effort to realize them in suitable mathematical settings where their 'non-standard' assumptions are obeyed. Some major achievements of mathematical logic in the first half of the twentieth century, such as the Gödel Completeness and Compactness Theorems in 1930, have model-theoretic traces. However it seems that it was only in 1954 that the name "model theory", more precisely "theory of models", was explicitly used for the first time, by Alfred Tarski. As a matter of fact in the 1950s, thanks to the work of Tarski himself, Abraham Robinson, Vaught, and others, model theory started to emerge as a new independent topic in mathematical logic.

Today model theory has a sophisticated and currently evolving internal abstract theory, together with vital outward-facing interactions with other branches of mathematics, including parts of algebra, combinatorics, geometry, number theory and analysis, often leading to striking applications. These twin features are often labelled ‘pure model theory' and ‘applied model theory'. In the 1970s and early 1980s they were sometimes pursued rather independently, but the pure and applied aspects are now closely intertwined, feeding off and stimulating each other.

The specific topics that will be considered in this special session are the following:

- Internal theory
- (Quasi) minimal structures and exponential fields
- O-minimal structures and groups definable in them
- Exponentiation and Lie algebras
- Weak Arithmetic
- Spain-Italy cooperations

## Organisers

- Enrique Casanovas (University of Barcelona, Spain) - ecasanovas@ub.edu
- Carlo Toffalori (University of Camerino, Italy)

## Abstracts and schedule

Below you can download the schedule and the abstracts of all talks of this special session.

## Speakers

- Elías Baro (Complutense University of Madrid, Spain)

* Cartan subgroups of groups definable in o-minimal structures*

- Andrés Cordón (University of Sevilla, Spain)

* Existentially closed models in the framework of first order arithmetic*

- Paola D'Aquino (Second University of Naples, Italy)

* Recursively saturated real closed fields*

- Antongiulio Fornasiero (Second University of Naples, Italy)

* Topology in definably complete structures*

- Sonia L'Innocente (University of Camerino)

* Rings of definable scalars of some 3-modules*

- Vincenzo Mantova (SNS Pisa – University of Camerino, Italy)

* Raising to powers on the unit circle*

- Daniel Palacín (University of Münster, Germany)

* On superstable expansions of free abelian groups*

- Françoise Point (F. N. R. S - F. R. S, Belgium)

* Separably closed fields and contractive Ore modules*

- Giuseppina Terzo (Second University of Naples, Italy)

* Zero sets of exponential polynomials*

- Frank O. Wagner (University Claude Bernard Lyon 1, France)

* A Fitting Theorem for simple theories*