|Wednesday, December 3|
|Artem Chernikov, University of Paris 7, France||Applications of model theory to geometric Ramsey theory (see abstract)|
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.
The result is similar to Tao's recent algebraic regularity lemma, but covers an orthogonal class of examples (and applies in particular to definable graphs in o-minimal theories and in p-adics).
This is joint work with Sergei Starchenko.