Teaching plan for the course unit

 

 

Close imatge de maquetació

 

Print

 

General information

 

Course unit name: Advanced Model Theory

Course unit code: 569075

Academic year: 2021-2022

Coordinator: Enrique Casanovas Ruiz Fornells

Department: Department of Mathematics and Computer Science

Credits: 5

Single program: S

 

 

Estimated learning time

Total number of hours 125

 

Face-to-face and/or online activities

42

 

-  Lecture with practical component

Face-to-face

 

42

Supervised project

18

Independent learning

65

 

 

Recommendations

 


Further recommendations


It is very recommendable to have followed with success  a basic model theory course.

 

 

Competences to be gained during study

 

To be able to carry out independent study of standard textbooks  and journal articles on model theory.

 

To understand the interest and potential impact of model theory and its applications to mathematics.

 

To be able to handle standard model-theoretic problems, formulating them with accuracy  and knowing the usual techniques to solve them.

 

 

 

 

 

 

Learning objectives

 

Referring to knowledge


A deep knowledge of the basic notions, tools and results from model theory (saturation, definability, prime models, indiscernibles, etc).
Introduction to the elementary notions of stability theory (Morley and Baldwin-Lachlan theorems)
Familiarity with examples of complete first-order theories, particularly vector spaces and algebraically closed fields.

 

Referring to abilities, skills


To be able to prove completeness, omega-categoricity and quantifier elimination for given examples of first-order theories.
To characterize definable sets in models of a given theory.
To classify the complexity of a theory from the point of view of stability theory

 

 

Teaching blocks

 

1. PRELIMINARIES

*  Back and forth, saturation, the monster model, quantifier elimination, model-completeness,  and omega-categoricity.

2. DEFINABILITY

*  
Definable, type-definable and invariant relations.  Beth and Svenonious theorems. Definable and algebraic closure.  Imaginaries.

3. PREGEOMETRIES

*  
Closure operators with exchange. Independence, basis and dimension. Modularity. Vector spaces.

4. RANKS

*  
Cantor-Bendixson rank. Morley rank.  Omega-stability and superstability. Strongly minimal sets. Algebraically closed fields.

5. PRIME MODELS

*  Omitting types. Prime and atomic models. Indiscernibles.  Two cardinal theorems.

6. CATEGORICITY

*  
Morley and Baldwin-Lachlan’s theorems.

 

 

Teaching methods and general organization

 


Detailed exposition in class, with examples and full proofs.
Asking questions and proposing problems to be solved by the student.
Class discussions on the main notions, examples, statements and relevant proofs.
Presentation of a topic by a student, previously discussed with the instructor.

 

 

Official assessment of learning outcomes

 

Weekly exercises. 40 %

Oral presentation of a topic 10 % 

Two partial exams  25 % each.

 

Examination-based assessment

Final examination

 

 

Reading and study resources

Consulteu la disponibilitat a CERCABIB

Book

E. Casanovas  Advanced Model Theory.

  These Lecture Notes will be delivered during the course.

E. Casanovas.  Teoría de Modelos (Lecture Notes).  1999.

  Available at http://www.ub.es/modeltheory/casanovas

C.C. Chang and H.J. Keisler.  Model Theory. North Holland PC,  3rd ed. 1990.

W. Hodges. Model Theory.  Cambridge UP 1993.

D. Lascar. Stability in Model Theory. Longman Sci. and Tech.  1987.

D. Lascar. La théorie des modèles en peu de maux. Cassini, Paris 2009.

D. Marker. Model Theory: an introduction. Springer 2002.

D. Marker, M. Messmer and A. Pillay. Introduction to the Model Theory of Fields.  Lecture Notes in Logic 5. Springer 1996.

A. Pillay. An introduction to Stability Theory.  Oxford UP  1983.

A. Pillay. Geometrical Stability Theory.  Oxford UP. 1996.

B. Poizat.  Cours de théorie des modèles. Offilib. Villeurbanne 1985.

K. Tent and M. Ziegler. A course in Model Theory.  Lecture Notes in Logic 40. Cambridge UP 2012.