Southern Wisconsin Logic Colloquium

SWLC Schedule

Refreshments will be served in the 9th floor lounge a half hour before talks. Please check the schedule below. All talks will be in 901 Van Vleck Hall unless stated otherwise.

Date Time Speaker Title Cookies,
dinner, etc.
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
at 3:30/
dinner at 6
Tuesday, March 10 4:00 p.m. Uri Andrews, UW TBA cookies/juice
at 3:30
Tuesday, March 17 4:00 p.m. TBA TBA cookies/juice
at 3:30
Tuesday, April 7 4:00 p.m. TBA TBA cookies/juice
at 3:30
Tuesday, April 14 4:00 p.m. TBA TBA cookies/juice
at 3:30
Tuesday, April 21 4:00 p.m. Ellen Chih, University of California-Berkeley TBA cookies/juice
at 3:30/
dinner at 6
Tuesday, April 28 4:00 p.m. Iván Ongay Valverde, UW TBA (specialty exam) cookies/juice
at 3:30
Tuesday, May 5 4:00 p.m. Ethan McCarthy, UW TBA (specialty exam) cookies/juice
at 3:30

Math 873 - Spring 2015 - Topics in Logic - The Descriptive Set Theory of Group Actions and Equivalence Relations

Instructor: Howard Becker

Prerequisites: None.

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.

Math 975 - Reading Seminar in Logic

Our reading seminar is meeting on Thursdays at 3:30 in B325 Van Vleck Hall.

Abstracts of talks

Baldwin's talk: The large and small in model theory: What are the amalgamation spectra of infinitary classes?

While the amalgamation and joint embedding properties (AP and JEP) for first order theories either hold in all cardinals or none, there is a cardinal dependence in sentences of Lω1 and more generally abstract elementary classes (AEC) that leads to problems on computing spectra of AP or JEP. The key tool for constructing examples is disjoint amalgamation. Here are some sample results:

Theorem 1 (Baldwin-Koerwien-Laskowski) There is a family of complete Lω1-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-sentence ψ

  1. whose models satisfy JEP(<λ0);
  2. that fails AP in all infinite cardinals;
  3. has 2λi+ many nonisomorphic maximal models in λi+, for all i ≤ α, but no maximal models in any other cardinality, while JEP fails for all larger cardinals; and
  4. has arbitrarily large models.

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.

Prepared by Steffen Lempp (">