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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.


Wednesday, January 9

  • Logic Seminar
    Andy Zucker (Université Paris Diderot)
      Bernoulli Disjointness
          2:00 -- 3:00 pm // Linde, Room 255

Abstract: 
We consider the concept of disjointness for topological dynamical systems, introduced by Furstenberg. We show that for every discrete group, every minimal flow is disjoint from the Bernoulli shift. We apply this to give a negative answer to the “Ellis problem” for all such groups. For countable groups, we show in addition that there exists a continuum-sized family of mutually disjoint free minimal systems. In the course of the proof, we also show that every countable ICC group admits a free minimal proximal flow, answering a question of Frisch, Tamuz, and Vahidi Ferdowsi.
(Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss)


Wednesday, January 16

  • Logic Seminar
    Forte Shinko (Caltech)
      Dense orbits in the space of subequivalence relations, I
          2:00 -- 3:00 pm // Linde, Room 255

Abstract:  Given a measure-preserving equivalence relation E, there is a Polish space S(E) of subequivalence relations, which admits a natural action of the full group [E]. Does S(E) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when E is the hyperfinite ergodic equivalence relation, and that the answer is no when E is induced by a measure-preserving action of a property (T) group.


Wednesday, January 23

  • Logic Seminar
    Forte Shinko (Caltech)
      Dense orbits in the space of subequivalence relations, II
          2:00 -- 3:00 pm // Linde, Room 255

Abstract:  Given a measure-preserving equivalence relation E, there is a Polish space S(E) of subequivalence relations, which admits a natural action of the full group [E]. Does S(E) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when E is the hyperfinite ergodic equivalence relation, and that the answer is no when E is induced by a measure-preserving action of a property (T) group.



Wednesday, February 6

  • Logic Seminar
    Forte Shinko (Caltech)
     Measures agreeing on invariant subsets, I
          2:00 -- 3:00 pm // Linde, Room 255



Wednesday, March 13

  • Logic Seminar
    Forte Shinko (Caltech)
     Measures agreeing on invariant subsets, II
          2:00 -- 3:00 pm // Linde, Room 255

Abstract:  We generalize the theorem of Þórisson, characterizing when two measures agree on invariant sets, to the setting of cardinal algebras.


Thursday, April 4

  • Logic Seminar
    Alexander Kechris (Caltech)
     Weak containment, co-induction and invariant random subgroups
          3:00 -- 4:00 pm // Linde, Room 255

Abstract:  We discuss the notion of weak containment and weak equivalence for pmp actions of countable groups and its relation with invariant random subgroups.

Thursday, April 11

  • Logic Seminar
    Todor Tsankov ( University of Paris)
      Bernoulli disjointness
          3:00 -- 4:00 pm // Linde, Room 255

Abstract:  The concept of disjointness of dynamical systems (both topological and
measure-theoretic) was introduced by Furstenberg in the 60s and has
since then become a fundamental tool in dynamics. In this talk, I will
discuss disjointness of topological systems of discrete groups. More
precisely, generalizing a theorem of Furstenberg (who proved the result
for the group of integers), we show that for any discrete group G, the
Bernoulli shift 2^G is disjoint from any minimal dynamical system. This
result, together with techniques of Furstenberg, some tools from the
theory of strongly irreducible subshifts, and Baire category methods,
allows us to answer several open questions in topological dynamics: we
solve the so-called "Ellis problem" for discrete groups and characterize
the underlying topological space for the universal minimal flow of
discrete groups. This is joint work with Eli Glasner, Benjamin Weiss,
and Andy Zucker.


Thursday, May 16

  • Logic Seminar
    Andrew Marks (UCLA)
      A characterization of \Sigma^0_{n+2}-hardness
          3:00 -- 4:00 pm // Linde, Room 255

Abstract:  We give a Baire category characterization of when a subset
of a Polish space is \Sigma^0_{n+2}-hard for n > 0. Our proof uses a
priority argument, and Antonio Montalban's true stages machinery. We
apply this characterization to the decomposability conjecture; the
problem of describing when a function is a union of countably many
continuous functions defined on \Pi^0_n sets.


Tuesday, May 28

  • Logic Seminar
    Martino Lupini (Victoria University of Wellington)
    Nonstandard analysis and Diophantine equations
          3:00 -- 4:00 pm // Linde, Room 255

Abstract:  I will give an overview of the application of nonstandard methods to the study of partition regularity of Diophantine equations. I will the explain how these methods can be used to generalize the classical Rado criterion for linear equations to obtain natural necessary conditions for arbitrary Diophantine equations, which are also sufficient for certain degree 2 equations. This is joint work with Jordan M. Barrett and Joel Moreira.

Tuesday, June 4

  • Logic Seminar
    Jeffrey Bergfalk (UNAM, Morelia Campus)
    Reformulated Ramsey relations and $\aleph_2$
          3:00 -- 4:00 pm // Linde, Room 255

Abstract


Thursday, June 13

  • Logic Seminar
    Slawek Solecki (Cornell University)
    Polishable equivalence relations
          3:00 -- 4:00 pm // Linde, Room 255

Abstract:  I will define a class of equivalence relations called Polishable equivalence relations that lies between the class of orbit equivalence relations of Polish group actions and the class of idealistic equivalence relations of Kechris and Louveau. I will present a Scott analysis for such equivalence relations. I will compare this analysis with the Scott analysis for isomorphism equivalence relations from continuous model theory and with versions of the Scott analysis for (certain) orbit equivalence relations of Polish group actions. As a tool in the proofs, I will introduce transfinite filtrations from one topology to another, a new notion of interpolation between topologies that may be of independent interest.





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

2013-2014

2014-2015

2015-2016

2016-2017

2017-2018