Rami Grossberg (Rami@cmu.edu)

URL: www.math.cmu.edu/~rami

MWF 3:30-4:20PM, WeH 7201

Starting date: Monday, January , 2015

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 2014.

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."

Text: There is no official text.

Some of the material appears in the following books:

- J. Baldwin, Fundamentals of stability theory.
- S. Buechler, Essential Stability theory.
- A. Pillay, Stability.
- Bruno Poizat, A course in Model Theory, Springer-Verlag 2000.
- Saharon Shelah, Classification Theory North-Holland 1990. This is the most important and difficult book in model theory. A free copy is available from his web site .
- Katrin Tent and Martin Ziegler, A course in model theory.

Rami's home page.

Last modified:
December 30 ^{nd}, 2014 |