Logic seminar, Spring 2013





Tuesday, 1/29: Isomorphism types of $\aleph_2$-dense subsets of $\mathbb{R}$, part I
  Justin Moore, Cornell University

Wednesday, 1/30: Introduction to countable Borel equivalence relations, part I
  Clinton Conley, Cornell University



Tuesday, 2/5: Isomorphism types of $\aleph_2$-dense subsets of $\mathbb{R}$, part II
  Justin Moore, Cornell University

Wednesday, 2/6: Countable Borel equivalence relations, part II
  Iian Smythe, Cornell University



Tuesday, 2/12: Epistemic game theory
  Adam Bjorndahl, Cornell University

Wednesday,2/13: Countable Borel equivalence relations, part III
  Iian Smythe, Cornell University



Tuesday, 2/19: Language-based games
  Adam Bjorndahl, Cornell University

Wednesday, 2/20: Countable abelian group actions and hyperfinite equivalence relations, part I
  Hossein Lamei Ramandi, Cornell University



Tuesday, 2/26: A norm for Tsirelson space
  Diana Ojeda Aristizabal, Cornell University

Wednesday, 2/27: Countable abelian group actions and hyperfinite equivalence relations, part II
  Hossein Lamei Ramandi, Cornell University



Tuesday, 3/5: Universal locally countable partial orders from recursion theory, and countable Borel equivalence relations
  Andrew Marks, Caltech

Wednesday, 3/6: Countable abelian group actions and hyperfinite equivalence relations, part III
  Jeffrey Bergfalk, Cornell University



Tuesday, 3/12: A norm for Tsirelson space, part II
  Diana Ojeda Aristizabal, Cornell University

Wednesday, 3/13: Group colorings and Bernoulli subflows
  Jeffrey Bergfalk, Cornell University



Tuesday, 3/26: Model theory within a subsystem of second-order arithmetic
  David Belanger, Cornell University

Wednesday, 3/27: Introduction to the theory of cost
  Diana Ojeda Aristizabal, Cornell University



Tuesday, 4/2: Model theory within a subsystem of second-order arithmetic, part II
  David Belanger, Cornell University

Wednesday, 4/3: Cost and treeable equivalence relations
  Adam Bjorndahl, Cornell University



Tuesday, 4/9: Countable locally nilpotent group actions and hyperfinite equivalence relations
  Brandon Seward, University of Michigan
A Borel equivalence relation is hyperfinite if it is the increasing union of Borel equivalence relations having finite classes. A long-standing open problem in descriptive set theory asks if Borel actions of countable amenable groups always induce (via their orbits) hyperfinite equivalence relations. In this talk I will discuss joint work with Scott Schneider which shows that this question has a positive answer for free Borel actions of countable locally nilpotent groups.

Wednesday, 4/10: Local complexity among treeable equivalence relations
  Clinton Conley, Cornell University
Group-theoretic rigidity techniques such as Zimmer and Popa cocycle superrigidity have been instrumental in works of Adams-Kechris, Thomas, and Hjorth (among others) in realizing complexity in the partial order of Borel reducibility among countable Borel equivalence relations. We introduce an elementary notion of rigidity which interacts better with Borel reducibility, allowing us to localize various complexity results to just above measure hyperfinite in the class of treeable equivalence relations. This is joint work with Ben Miller.



Tuesday, 4/16: Complicated residually finite groups
  Mark Sapir, Vanderbilt University
We give the first examples of computationally complicated residually finite finitely presented groups. This is a joint work with Olga Kharlampovich and Alexei Miasnikov.

Wednesday, 4/17: Cost and treeable equivalence relations, part II
  David Belanger, Cornell University



Tuesday, 4/23: Borel Complete Sections on Bernoulli Shifts
  Su Gao, University of North Texas
Constructions of layered Borel complete sections with regularity properties are important for hyperfiniteness proofs and general study of countable Borel equivalence relations. A classical theorem of Slaman-Steel gives the existence of vanishing layers of Borel complete sections. In this talk I prove a boundedness property for any layered Borel complete sections for the Bernoulli shift on Z. This implies that layered Borel complete sections with certain desirable properties do not exist. This is joint work with Steve Jackson and Brandon Seward.

Wednesday, 4/24: The Gaboriau-Lyons dynamic version of the von Neumann conjecture
  Scott Messick, Cornell University



Tuesday, 4/30: A Computability Theoretic equivalent to Vaught's Conjecture.
  Antonio Montalbán, UC Berkeley
We find two computability theoretic properties on the models of a theory T which hold if and only if T is a counterexample to Vaught's conjecture.

Wednesday, 5/1: The Gaboriau-Lyons dynamic version of the von Neumann conjecture, part II
  Scott Messick, Cornell University



Tuesday, 5/7: Algorithmic Randomness via Random Algorithms
  Mia Minnes, UC San Diego
Algorithmic randomness defines what it means for a single mathematical object to be random. This active area of computability theory has been particularly fruitful in the past several decades, both in terms of expanding theory and increasing interaction with other areas of math and computer science. Randomness can be equivalently understood in terms of measure theory, Kolmogorov complexity (incompressibility), and martingales.

In this context, we present a novel definition of betting strategies that uses probabilistic algorithms also studied in complexity theory. This definition leads to new characterizations of several central notions in algorithmic randomness and addresses Schnorr's critique, a longstanding philosophical question in algorithmic randomness. Moreover, these techniques have yielded new proofs of complicated separation theorems and suggest new approaches for tackling one of the biggest open questions in the field (KL = ML?). This is joint work with Sam Buss.