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 TBA |

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 TBA |

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 |

**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.

**Theorem 1** (Baldwin-Koerwien-Laskowski) There is a family of complete
L_{ω1,ω}-sentences φ_{n} for
1 ≤ n < ω such that φ_{n} characterizes
ℵ_{n} and all models in ℵ_{n} are maximal. The
class satisfies amalgamation in ℵ_{k} for k ≤ n-2, fails it
in ℵ_{n-1} and trivially satisfies it in
ℵ_{n}.

**Theorem 2** (Baldwin-Koerwien-Souldatos) If <λ_{i} |
i ≤ α ≤ ω> is any increasing sequence of cardinals below
ℵ_{ω}, then there exists an
L_{ω1,ω}-sentence ψ

- whose models satisfy JEP(<λ
_{0}); - that fails AP in all infinite cardinals;
- has 2
^{λi+}many nonisomorphic maximal models in λ_{i}^{+}, for all i ≤ α, but no maximal models in any other cardinality, while JEP fails for all larger cardinals; and - 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 (@math.wisc.edu">lemppmath.wisc.edu)