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Logic Seminar
2018-2019

 


Thursday, October 11

  • Logic Seminar
    Felix Weilacher (Caltech)
    Marked groups with isomorphic Cayley graphs but different Borel combinatorics
    3:00 -- 4:00 pm // Linde, Room 255


Abstract: We construct pairs of marked groups with isomorphic Cayley graphs but different Borel chromatic numbers for the free parts of their shift graphs. This answers a question of Kechris and Marks. We also show that these graphs have different Baire measurable and measure chromatic numbers, answering analogous versions of the question.


Thursday, October 25

  • Logic Seminar
    Assaf Shani (UCLA)
    Countable Borel equivalence relations and weak choice principles 
          3:00 -- 4:00 pm // Linde, Room 255

Abstract: For a countable Borel equivalence relation E we consider the weak choice principle ``every countable sequence of E-classes has a choice function''. We establish a relationship between ergodicity of the equivalence relations and the study of these choice principles. 
We will separate these choice principles as follows: if E is F-ergodic (with respect to some measure) then there is a model of set theory in which ``choice for E classes'' fails yet ``choice for F classes'' holds. For example, ``choice for E_\infty classes'' is strictly stronger than ``choice for E_0 classes''.

A key lemma in the proof is the following statement: if E is F-ergodic with respect to an E-quasi-invariant measure \mu then the countable power of E, E^\omega, is F-ergodic with respect to the product measure \mu^\omega. The proof relies on ideas from the study of weak choice principles.

In this talk we will go over some of the basic ideas behind forcing and explain how they were used to construct models of set theory without the axiom of choice. We will then establish the relationship with ergodicity and focus on proving the lemma mentioned above. 
  

Thursday, November 1

  • Logic Seminar
    Anush Tserunyan (University of Illinois, Urbana-Champaign)
      Hyperfinite subequivalence relations of treed equivalence relations, I
          3:00 -- 4:00 pm // Linde, Room 255

Abstract: A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed quasi-measure-preserving equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. Most importantly, just like every cyclic subgroup is contained in a unique maximal one, we show that every hyperfinite subequivalence relation is contained in a unique maximal one.


Thursday, November 8

  • Logic Seminar
    Anush Tserunyan (University of Illinois, Urbana-Champaign)
      Hyperfinite subequivalence relations of treed equivalence relations, II
          3:00 -- 4:00 pm // Linde, Room 255

Abstract: A large part of measured group theory studies structural properties of countable groups that hold "on average". This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on probability spaces. In this vein, the cyclic (more generally, amenable) groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed quasi-measure-preserving equivalence relation, deriving some of analogues of structural properties of cyclic subgroups of a free group. Most importantly, just like every cyclic subgroup is contained in a unique maximal one, we show that every hyperfinite subequivalence relation is contained in a unique maximal one.









Contact information: A. Kechris, kechris@caltech.edu
Previous seminars

2013-2014

2014-2015

2015-2016

2016-2017

2017-2018