Course Meeting:

- Mon May 17, Wed May 19
- Wed May 26, Thu May 27
- Mon May 31, Wed June 2, Thu June 3
- Thu June 10

Thursday lectures are 14:15-16:15 in salle 225-227.

Instructor: David Marker

e-mail: marker@math.uic.edu

course webpage: http://www.math.uic.edu/~marker/orsay

- Logic, Language and Structures
- The Compactness Theorem and applications
- Ultraproducts and a proof of compactness
- Ax's Theorem that injective polynomial maps are surjective
- Quantifer elimination tests
- the model theory of algebraically closed fields and algebraic geometry
- the model theory of real closed fields and semialgebraic geometry

- o-minimality, subanalytic geometry and exponentiation
- Asymptotic bounds on the number of rational points on o-minimal sets and Diophantine applications

- D. Marker,
*Model Theory: An Introduction*, Graduate Texts in Mathematics 217, Springer, New York, 2002.

- Sections 1--3: Language, Structures and Theories, The Compactness Theorem, Ultraproducts and Compactness
- Sections 4-6: Complete Theories, Quantifier Elimination, Algebraically Closed Fields
- Section 7: Real Closed Fields and o-minimality
- Section 8: The Pila-Zannier proof of the Manin-Mumford Conjecture These notes don't completely correspond to my lecture. They give the full proof of Pila-Zanier from Pila-Wilkie and don't contain the introductory remarks on abelian varieties and Manin-Mumford. For a survey of the introductory material see:
- Appendix: Real Algebra