Sebastien Vasey: Quasiminimal structures and excellence

 

Here is information about near  future model theory seminars.  If you are interested to speak or have suggestions for speakers please contact me.  Further information can be found on http://www.math.cmu.edu/math/modeltheoryseminars/modeltheoryseminars.php
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Model Theory Seminar
 
Sebastien Vasey
 
Carnegie Mellon University

Title:  Quasiminimal structures and excellence. Part 1

Abstract:  A recurring theme in model theory is upward transfer of a property, using only information about countable structures to derive information on all uncountable structures. For example, knowing that an Abstract Elementary Class has no maximal countable model lets us deduce it has an uncountable model.  A generalization of this fact was proved by Shelah in a milestone paper, where he showed how to obtain existence of models of an $L_{\omega_1, \omega}$ sentence at $\aleph_n$ using a property of systems of countable models index by $n$-dimensional cubes that he called $n$-goodness. A class where $n$-goodness holds for all $n$ is called excellent. It turns out that excellence implies existence of arbitrarily large models, and much more.

The concept of excellence got renewed attention, when Boris Zilber used it to study Schanuel’s conjecture for the complex numbers. Zilber studied quasiminimal classes that had to satisfy several requirements, including a version of excellence. Until recently, it was unclear whether the excellence requirement was necessary. In this series of talks, I will present a result of Bays, Hart, Hyttinen, Kesälä and Kirby that shows it is not: Excellence holds for free in quasiminimal classes. I aim to make the talks reasonably self-contained and use only minimal backgroud.

Date: Monday, October 14, 2013
Time: 5:00 pm
Location: Wean 8220
=================================
Model Theory Seminar
 
Sebastien Vasey
 
Carnegie Mellon University

Title:  Title: Quasiminimal structures and excellence. Part 2

Abstract:  A recurring theme in model theory is upward transfer of a property, using only information about countable structures to derive information on all uncountable structures. For example, knowing that an Abstract Elementary Class has no maximal countable model lets us deduce it has an uncountable model.  A generalization of this fact was proved by Shelah in a milestone paper, where he showed how to obtain existence of models of an $L_{\omega_1, \omega}$ sentence at $\aleph_n$ using a property of systems of countable models index by $n$-dimensional cubes that he called $n$-goodness. A class where $n$-goodness holds for all $n$ is called excellent. It turns out that excellence implies existence of arbitrarily large models, and much more.

The concept of excellence got renewed attention, when Boris Zilber used it to study Schanuel’s conjecture for the complex numbers. Zilber studied quasiminimal classes that had to satisfy several requirements, including a version of excellence. Until recently, it was unclear whether the excellence requirement was necessary. In this series of talks, I will present a result of Bays, Hart, Hyttinen, Kesälä and Kirby that shows it is not: Excellence holds for free in quasiminimal classes. I aim to make the talks reasonably self-contained and use only minimal backgroud.

Date: Monday, October 21, 2013
Time: 5:00 pm
Location: Wean 8220

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