Andres Villaveces: Model Theory in Abstract Elementary Classes

This month, Andres Villaveces will visit Iran and will give a series of lectures in Institute for Research in Fundamental Sciences (IPM) and Amirkabir University of Technology (AUT).
Here is the abstract of his talks according the last announcement:

Title: Model Theory in Abstract Elementary Classes


Stability theory for first order logic developed very quickly after 1970, prompted by the work of Morley, Lachlan, Baldwin, Shelah and Lascar, among others. Earlier on, model theory of infinitary logics, generalized quantifiers had been started (Keisler, Mostowski) but its stability theory was for a long time postponed. Around 25 years ago, a synthesis of many of these logics started, with semantic rather than syntactic emphasis, with focus on the classes of models rather than on the logics axiomatizing them. This synthesis, started by Shelah, is Abstract Elementary Classes. What followed was the development of a very rich and structural stability theory for these classes. Recently, the emphasis has gone back to the interplay with many logics, on the one hand, and with category theory, on the other hand.

There will be four sessions, organized as follows:

1- Sunday 22th Nov. 10-12 in IPM

The Basics: definitions of AECs, examples, the Presentation Theorem. Galois types, stability, independence notions (splitting), the categoricity conjecture and partial results.

2- Tuesday 24th Nov.13-15 in AUT

A proof of categoricity transfer: this part of the minicourse will have a sketch of a proof of categoricity transfer (Vaughtian pairs, rooted types).
3- Thursday 26th Nov. 14-16 in IPM.

Categoricity and Large Cardinals. More independence notions (connected to splitting, forking), tameness and large cardinals. The Categoricity Conjecture (consistency).

4- Sunday 29th Nov. 10-12 in IPM.

Category theoretic versions. This will be an exploration of recent category theoretic constructions connected with the model theory of AECs. This will also be a summary of the previous lectures, done through the adaptation to category theory.

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