21-803 Model Theory III
Rami Grossberg (Rami@cmu.edu)
URL: www.math.cmu.edu/~rami
MWF 3:30-4:20PM, WeH 7201
Starting date: Monday, January , 2017
12 Units

This will be different than courses I offered in the last two years. It will not depend on Model theory II offered in spring 2016.

I will concentrate in classification theory for first-order theories. The theory was developed mostly by Saharon Shelah presented in his 1990 book and in several hundreds of papers. I will present a modern overview of Shelah's theory incorporating few recent innovations and simplifications. The development of the theory was motivated by set-theoretic questions like: "what is the asymptotic behavior of the function I(\aleph_\alpha,T) as a function of \alpha ?" and "what is the first \lambda such that an uncountable first-order stable theory T is stable in \lambda?"

Surprisingly the full answer to such combinatorial set-theoretic questions led for a development and discovery of a conceptually rich theory which seems to be related to aspects of commutative algebra and algebraic-geometry. This theory found several applications in the form of solving fundamental problems of classical fields of mathematics among them geometry and number theory.

The focus will be on the simplest and most fundamental aspects of the pure theory. Primarily around a notion called forking and various characterizations of classes of theories.

The American Mathematical Society awarded in 2013 its "Steel prize for Seminal Contribution to Research" to Shelah for his 1990 book. [the prize is given out to logicians once in 10 years, the 2003 prize was shared by Jensen & Morley]
Among the reasons given: ... made model theory into a mature field, completely transforming its aims, methods, and ability to connect to algebra and geometry."

Prerequisites: Elementary model theory (21-603) or permission of the instructor. The pace of presentation will be faster than in 21-603.

Text: There is no official text.

Some of the material appears in the following books:

Rami's home page.
Last modified: January 18 nd, 2017