MATH 7810: Logic Seminar: set theoretic methods in analysis (fall 2008)

Instructor: Justin Moore

This semester the logic seminar will explore applications of set theory and logic to other areas of mathematics. The aim of the course is in part to engage people who may have had little exposure to logic and hence an effort will be made to cover any relevant background material. Students should be comfortable with basic concepts from algebra, functional analysis, and topology. The exact content will be determined based on the interests of the participants, but the following is a tentative list of topics (further suggestions are welcome):

• Definable equivalence relations and their relationship to topological group actions and ergodic theory. This will encompass both the dichotomy theorems for Borel equivalence relations (whose proofs involve methods from recursion theory) as well as applications such as those found in the work of Hjorth, Kechris, Louveau, Solecki, and others. See Kechris's article New directions in descriptive set theory in the Bulletin of Symbolic Logic for more information.
• Extreme amenability of topological groups: A topological group is extremely amenable if any action on a compact space has a fixed point. Kechris, Pestov, and Todorcevic have results connecting extreme amenability of automorphism groups to structural Ramsey theory and it seems that this is only the beginning of an important interaction between Ramsey theory, infinite dimensional geometry, and topological dynamics. See Pestov's recent AMS book for more information.
• C*-algebras and set theory. Recently Phillips-Weaver and Farah have shown that the assertion "The Calkin algebra has only inner automorphisms" is independent of the usual axioms of set theory. Here the Calkin algebra is the collection of bounded operators on $\ell_2$ modulo the compact operators. This has sparked what appears to be a fertile interaction between set theorists and C*-algebraists. See Nik Weaker's article Set theory and C*-algebras in the Bulletin of Symbolic Logic.

Traditionally the seminar meets twice a week (Tuesday and Wednesday afternoon) with Tuesday devoted to a specific topic and Wednesday devoted to a selection of topics throughout logic and to presentations by outside speakers. This semester an additional alternate meeting time will be devoted to a second applications topic. Students enrolled in the course will be expected to attend the seminar twice a week and to take turns presenting material.