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.
Wednesday, December 3
(dept. colloquium)
4:00 p.m.
(room B239)
Artem Chernikov, University of Paris 7, France Applications of model theory to geometric Ramsey theory (see abstract)  

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 Wednesdays at 3:30 in room TBA.

Abstracts of talks

Chernikov's talk: Applications of model theory to geometric Ramsey theory

In a series of papers by Alon, Conlon, Fox, Gromov, Naor, Pach, Pinchasi, Radoičić, Sharir, Sudakov, Lafforgue, Suk and others, it was demonstrated that families of graphs with the edge relation given by a semialgebraic relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces modulo a small mistake (for example, the incidence relation between points and lines on the real plane, or higher dimensional analogues). We show that in fact the whole theory can be developed for families of graphs whose edge relation is uniformly definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of measures. Moreover, distality characterizes these strong regularity properties.

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.


Prepared by Steffen Lempp (@math.wisc.edu">lemppmath.wisc.edu)