Recursion Theory/Descriptive Set Theory Seminar - Fall 2024

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Time: Fridays 11:00-12:30
Location: 736 Evans

Topic: Martin's conjecture and countable Borel equivalence relations.

Tentative Schedule:

• Sep 6 - 13: Sean and Patrick. Martin measure and its basic properties, Martin's conjecture. Steel's games and theorems on uniformly invarinat functions
  • Martin (1968) The axiom of determinateness and reduction principles in the analytical hierarchy
  • Steel (1982) A classification of jump operators
• Sep 20-27: Robert. Finishing the proof that Martin's conjecture is true for uniformly invariant functions. If f(x) < x, then f is constant on a cone. Martin's conjecture for increasing, order-preserving functions. Turing equivalence is not hyperfinite. Bard's theorem on uniformly invariant functions.
  • Slaman-Steel (1988) Definable functions on degrees
  • Bard (2020) Uniform Martin's conjecture, Locally
• Oct 4: Yiping:
  • Lutz, Siskind (2024) Part 1 of Martin's conjecture for order-preserving and measure-preserving functions
• Oct 11: Liang Yu: When A + xA = R
• Oct 18: Su Gao: Borel class-wise Z-orderings and hyperfiniteness
• Oct 25: Yvette: Introduction to Martin's conjecture and countable Borel equivalence relations
  • Thomas, Schneider (2013) Countable Borel Equivalence Relations
  • Dougherty, Kechris (2000) How many Turing Degrees are there?
• Nov 1, 8: Katalin: Introduction to property (T) groups
  • Bekka, de la Harpe, Valette (2008) Kazhdan's Property (T)
• Nov 15, 22: Alex K.: Ioana's rigidity theorem for property (T) groups.
  • Ioana (2011) Cocycle superrigidity for profinite actions of property (T) Groups"
• Dec 6: Alex T.: Applications of superrigidity to countable Borel equivalence relations.
  • Thomas (2009) Popa Superrigidity and countable Borel equivalence relations
• Dec 13: Liza: Martin's conjecture, CBERs, and superrigidity
  • Thomas (2009) Martin's conjecture and strong ergodicity
  • Thomas (2012) Universal Borel actions of countable groups

Some other related papers:

• Furman (2007) On Popa's Cocycle Superrigidity Theorem
• Thomas (2002) Some applications of superridigity to Borel equivalence relations
• Coskey (2013) Ioana's superrigidity theorem and orbit equivalence relations
• Marks, Slaman, Steel (2016) Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations
• Kihara, Montalbán (2018) The uniform Martin's conjecture for many-one degrees
• Montalbán (2019) Martin's conjecture: A classification of the naturally occuring Turing degrees
• Day, Marks (2020) On a question of Slaman and Steel
• Lutz (2021) Results on Martin's conjecture
• Lutz (2024) Martin's conjecture for regressive functions on the hyperarithmetic degrees
• Frisch, Kechris, Shinko, Vidnyánszky (2023) Realizations of countable Borel equivalence relations