Series: Penn State Logic Seminar

Date: Tuesday, January 28, 2003

Time: 2:30 - 3:45 PM

Place: 113 McAllister Building

Speaker: Alexei Kolesnikov, CMU, Mathematics

Title: n-Simple Theories and Generalized Amalgamation

Abstract:

Simple theories were introduced by Saharon Shelah in 1978 as a part of
his program to find meaningful dividing lines in unstable first order
theories. Following the work of Byunghan Kim (1996), simple theories
were extensively studied by Steve Buechler, Bradd Hart, Anand Pillay,
Frank Wagner, and many others. One of the reasons for such interest
was Ehud Hrushovski's earlier work on difference fields, which falls
into the framework of simple theories.

Shelah in his list of open problems in models theory [Sh702]
conjectured existence of $\omega+1$ subclasses in the class of all
simple theories. I will present my solution to the conjecture.
Surprisingly, the motivation for this solution comes from Shelah's
work on classification theory for non-elementary classes, more
specifically, the model theory of excellent classes. The latter too
came in focus recently due to Boris Zilber's work on fields with
pseudoexponentiation.

I will introduce the necessary background, state the new definitions
and results, and will talk about the non-first order motivation behind
them.

REFERENCES: [Sh702] Saharon Shelah. On what I do not understand (and
have something to say), model theory. Mathematica Japonica, 2000.
Vol. 51, No. 2, 329-378.