**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 more than 30 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 also 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 about a thousand pages of 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
decade 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 a much greater impact.

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 theÊstudy of classes of finite models, butÊI 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 focus on the basic parts of the theory that 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/mt2.10.desc.html

Rami's home page.

Last modified:
November ^{10th}, 2010 |