|Tuesday, March 3||4:00 p.m.||John Baldwin, University of Illinois at Chicago||The large and small in model theory: What are the amalgamation spectra of infinitary classes? (see abstract)||cookies/juice|
dinner at 6
|Tuesday, March 10||4:00 p.m.||Uri Andrews, UW||TBA||cookies/juice|
|Tuesday, March 17||4:00 p.m.||TBA||TBA||cookies/juice|
|Tuesday, April 7||4:00 p.m.||TBA||TBA||cookies/juice|
|Tuesday, April 14||4:00 p.m.||TBA||TBA||cookies/juice|
|Tuesday, April 21||4:00 p.m.||Ellen Chih, University of California-Berkeley||TBA||cookies/juice|
dinner at 6
|Tuesday, April 28||4:00 p.m.||Iván Ongay Valverde, UW||TBA (specialty exam)||cookies/juice|
|Tuesday, May 5||4:00 p.m.||Ethan McCarthy, UW||TBA (specialty exam)||cookies/juice|
Time and Place: MWF 13:20-14:10
Recommended Textbook: Su Gao: Invariant Descriptive Set Theory
Course Description: Descriptive set theory is the study of definable (e.g., Borel, analytic, etc.) subsets of Polish spaces. We consider two topics in connection with -- or from the point of view of -- descriptive set theory. One is definable (e.g., continuous or Borel) actions of Polish groups. A special case of this is the logic actions, where the orbit equivalence relation is isomorphism; therefore, to some extent, this subject is a generalization of the model theory of infinitary languages and countable structures.
The other topic is definable equivalence relations. While the two topics are closely related, neither is a subtopic of the other, since there is more to a group action than the orbit equivalence relation and there are some very simple equivalence relations that cannot be realized as orbit equivalence relations. There is a partial ordering on equivalence relations called Borel reducibility. One interpretation of this is that E is Borel-reducible to F means that the "Borel cardinality" of the equivalence classes of E is less than or equal to that of F. Another interpretation has to do with classification: It means that classifying the F-equivalence classes by Borel invariants is at least as difficult as classifying the E-equivalence classes. The subject matter of this course derives from diverse sources, including logic, ergodic theory, operator algebras and representation theory. It has applications to all of these fields.
The student should have some experience -- but need not have much experience -- with classical and effective descriptive set theory.
Theorem 1 (Baldwin-Koerwien-Laskowski) There is a family of complete Lω1,ω-sentences φn for 1 ≤ n < ω such that φn characterizes ℵn and all models in ℵn are maximal. The class satisfies amalgamation in ℵk for k ≤ n-2, fails it in ℵn-1 and trivially satisfies it in ℵn.
Theorem 2 (Baldwin-Koerwien-Souldatos) If <λi | i ≤ α ≤ ω> is any increasing sequence of cardinals below ℵω, then there exists an Lω1,ω-sentence ψ
On the other hand, there are upper bounds for the Hanf number of AP or JEP:
Theorem 3 (Baldwin-Boney) If there is strongly compact cardinal κ and AP (JEP) holds for an AEC on a tail of κ then it holds everywhere above κ.