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.