## Course on Model theory, Utrecht, Fall 2012

Contents

The aim of the course is to give an introduction to modern model theory, especially stability theory. Stability theory was first developed (mainly by Shelah) as a part of pure model theory. In the hands of Zil'ber and Hrushovski it has become increasingly relevant and applicable to other parts of mathematics, especially algebraic geometry. The most striking application of this kind was Hrushovski's proof in 1996 of the Mordell-Lang Conjecture. We will cover the basics of stability theory and prove Morley's Theorem from 1965, which was the starting point for the whole development.

Prerequisites

A background in logic on the level of the local "Foundations of Mathematics" course. See the lecture notes "Sets, Models and Proofs" by I. Moerdijk and J. van Oosten (http://www.staff.science.uu.nl/~ooste110/syllabi/setsproofs09.pdf).

Some basic knowledge of topology and algebra (rings, fields, vector spaces) would be good, in order to be able to appreciate the examples.

Practical details

Period: week 37 - 51

Time: Friday 11:00 - 13:00

First lecture: September 14

Location: Minnaert building 204

There will be no lecture on November 9. An additional lecture is scheduled for Monday November 19, 11:00-13:00 in BBL 077.

From November 16 onwards the lectures will take place on Fridays, 9:00-11:00 in room 611 in the Math Building.

The final lecture on December 23 will take place in the Minnaert Building, Room 023.

Lectures

September 14: basic notions, universal theories, Skolem's theorem slides

September 21: Skolem theories, downward Löwenheim-Skolem theorem, compactness theorem, diagrams slides

September 28: Upward Löwenheim-Skolem theorem, directed systems, Robinson's Consistency Theorem slides

October 5: Craig interpolation, Beth's definability theorem, Chang-Los-Suszko theorem slides

October 12: Types, type spaces, saturated models slides

October 19: Universal, homogeneous, strongly homogeneous models, cofinality, regular and singular cardinals slides

October 26: Existence saturated models, Svenonius' theorem, omitting types theorem slides

November 2: Omega-categoricity, small theories, prime models slides

November 16: Small theories have prime models, examples: dense linear orders, Boolean algebras slides

November 19: Examples: Atomless Boolean algebras, vector spaces, algebraically closed fields

November 23: Examples: Algebraically closed fields finished, ordered fields

November 30: Examples: Real closed ordered fields, Hilbert's seventeenth problem handout

December 7: Stability, Ramsey's Theorem, order indiscernibles, uncountably categorical theories are omega-stable slides

December 14: Morley rang, Morley degree, totally transcendental theories slides

December 21: Constructible and atomic extensions, Morley's theorem slides

Complete set of slides

Take-home exam

Solutions for the take-home exam

Lecturer

Benno van den Berg

Email: B.vandenBerg1 AT uu.nl

Room: 502 in the Math Building

Homepage: can be found here.