Piotr Kowalski: Model Theory

This is a 22-lecture course, with each lecture being about 45 minutes or so, given online by Piotr Kowalski. It gives an introduction to model theory.

We state several classical results about fields (Ax’s theorem, Hilbert’s Nullstellensatz), which have easy model-theoretic proofs and then introduce the necessary basic model-theoretic tools to describe those proofs. In this way we motivate and present basic definitions and results from model theory. We will also sketch at the end of the lecture some recent applications of model theory regarding non-existence of classical solutions of some differential equations, which was a problem considered by Painlevé in 1895 and an argument was found only recently.

1. Ax’s Theorem
2. Hilbert’s Nullstellensatz
3. Ax’s Theorem and Nullstellensatz
4. Languages and Structures
5. Compactness Theorem I
6. Compactness Theorem II
7. Compactness Theorem: Proof
8. Compactness Theorem: Consequences
9. Lefschetz Property I
10. Lefschetz Property II
11. Quantifier Elimination I
12. Quantifier Elimination II
13. General Theory of Model Companions: Examples Part I
14. General Theory of Model Companions: Examples Part II
15. Existentially Closed Groups
16. Existentially Closed Rings
17. Infinite Galois Theory I
18. Infinite Galois Theory II
19. Infinite Galois Correspondence
20. First Properties of Existentially Closed G-Fields
21. G-Closed Fields and Frattini Covers
22. Model Companion of the Theory of G-Fields

These videos are of a lecture course by Piotr Kowalski at Nesin Mathematics Village (Izmir, Turkey) in 2021, and produced by CIMPA.