Instructor: Rami Grossberg
Office: WEH 7204
Phone: x8482 (268-8482 from external lines), messages at x2545
Email: Rami@cmu.edu
URL: www.math.cmu.edu/~rami
Office Hours: By appointment or whenever else you can find me.

Purpose. This is a second course in model theory. This semester the course will be entirely dedicated to "Classification Theory for Abstract Elementary Classes (AECs)."

Course description.

The subject started about 40 years ago by Shelah, the goal is to discover and introduce essentially category-theoretic concepts and tools sufficient for the development of model theory for various infinatry logics and ultimately to have a complete theory of invariants of models up to isomorphism in all cardinals, whenever this is possible and also establish the reason for nonexistence of a theory of invariants. Shelah proposed a conjecture as a test for the development of the theory: Shelah's categoricity conjecture, it is a parallel to Morley's categoriicty theorem for L_w1,w. Despite the existence of several of thousands of pages containing partial results the conjecture is still open. The analogue for classes of models of complete first-order theories is a highly developed theory called classification theory. In the last 20 years several very substantial applications of this theory to algebra, geometry and number theory were discovered. It is expected that eventually classification Theory for AECs will have great impact on fields of mathematics outside of logic.

Shelah in his list of open problems in model theory [Sh 702] writes: ``I see this [classification of Abstract Elementary Classes] as the major problem of model theory.''

Until 2001 virtually nobody besides Shelah published work on AECs, this is in part due to the highly complex nature of the original papers that not only introduce many new model-theoretic tools but also required relatively sophisticated set-theoretic considerations. Interest and progress in the field by others materialized from two directions: Boris Zilber managed to construct a function over the complex numbers sharing many formal properties with exponentiation and as well satisfying Schanuel's conjecture over the complex numbers (this is a far reaching conjecture in transcendental number theory implying solutions to many difficult long standing problems, e.g. it implies that \pi + \e is a transcendental number). Zilber's construction uses a combination of methods from number theory with abstract model-theoretic concepts of Shelah. In parallel Grossberg and VanDieren introduced the notion of tameness for AECs and proved new cases of Shelah's categoricity conjecture. Both directions influenced new people to enter the field.

Hopefully some of the techniques will turn to be useful also in the study of classes of finite models, I will discuss uncountable models only. I will focus on the parts of the theory that in my opinion are most likely to lead to new significant results.

I will emphasize the basic parts of the theory that in my opinion may eventually converge to a proof of cases of Shelah's categoricity conjecture and the existence of a model of cardinality \lambda^{++}. This involves among other things development of stability-like theory for AECs.

Recently three important books on the field were published:
(1) John T. Baldwin. Categoricity, AMS 2009 (235 pages).
(2) Saharon Shelah. Classification Theory for Abstract Elementary Classes Volume 1, College Publications 2009 (824 pages).
(3) Saharon Shelah. Classification Theory for Abstract Elementary Classes Volume 2, College Publications 2009 (702 pages).

Textbooks:

(1) Rami Grosberg. A course in Model Theory I: An Introduction .

(2) Rami Grosberg. A course in Model Theory III: Classification Theory for Abstract Elementary Classes.

My books are often revised, the most recent versions are available to students in my course from a password protected directory here. If you use this link you agree not to forward and/or share the contents without my explicit agreement.

Prerequisites: The contents of a basic graduate course in model theory like 21-603 or permission of the instructor.

Evaluation: Weekly homework assignments (20%) 30%, Midterm 20% and an in class 3 hour final 50%.

Course web page: http://www.math.cmu.edu/~rami/mt2.16.desc.html