Math 574, Topics in Logic and Foundations

Instructor: John Clemens

email: clemens@math.psu.edu

Homework:

1. Homework #1 (due 9-26-03): Postscript or PDF
2. Homework #2 (due 10-13-03): Postscript or PDF
3. Homework #3 (due 11-21-03): Postscript or PDF

Suggested References:

• Kechris, Classical Descriptive Set Theory
• Becker and Kechris, The Descriptive Set Theory of Polish Group Actions
• Hjorth, Classification and Orbit Equivalence Relations

Tentative Syllabus:

1. Introduction
   • What is Descriptive Set Theory?
   • What is a definable equivalence relation?
   • The definable cardinality of a quotient space
   • Using DST to compare the complexity of classification problems
2. Polish spaces
   • Properties of seperable, complete metric spaces
   • The Borel hierarchy and other definable sets
   • The Perfect Set Theorem for analytic sets
3. Equivalence relations
   • Definable equivalence relations on Polish spaces
   • Borel reducibility: comparing quotient spaces
   • The simplest relations: concrete classifiability
   • Pathologies without definability assumptions
4. Two Dichotomy theorems
   • Silver's Theorem: an analogue of the Perfect Set Theorem for co-analytic relations
   • Burgess's Theorem for analytic relations
   • Effective descriptive set theory
   • The Glimm-Effros Dichotomy for Borel equivalence relations
   • More equivalences of concrete classifiability
5. Hyperfinite equivalence relations
   • The simplest non-concretely classifiable relations
   • Invariant measures and compressibility
   • Classification up to bireducibility and biembedibility
6. Hypersmooth equivalence relations
   • Moving beyond countable relations
   • Essentially hyperfinite relations
   • A dichotomy for being essentially hyperfinite
7. Global properties of the reducibilty ordering
   • Embedding inclusion mod finite in the reducibility order
   • Jump-like operations
8. Countable Borel equivalence relations
   • Universal countable Borel equivalence relations
   • Treeable equivalence relations
   • Amenability
   • Actions of groups of polynomial growth
9. Actions of Polish groups
   • Orbit equivalence relations>
   • Universal groups and actions
   • The Topological Vaught's Conjecture
10. Classes of countable structures
    • Logical actions
    • The infinite symmetric group
    • Classification by countable structures
    • Hjorth's theorey of turbulence
11. Classification problems