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Logic Seminar
2017-2018

 


Tuesday, October 3

  • Logic Seminar
    Forte Shinko (Caltech)
    Hyperfiniteness of boundary actions of cubulated hyperbolic groups
    3:00 -- 4:00 pm // Mathematics, Room 131


Abstract: A classical result of Dougherty, Jackson and Kechris implies that the action of the free group on its Gromov boundary induces a hyperfinite equivalence relation. We will discuss a generalization of this result to a wider class of hyperbolic groups. Joint with Jingyin Huang and Marcin Sabok.


Tuesday, October 17

  • Logic Seminar
    Jefferey Bergfalk (Cornell)
    Higher walks and stronger homology theories
    3:00 -- 4:00 pm // Mathematics, Room 131



Abstract: We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Cech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the nth infinite cardinal, for each n, holding in ZFC, (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.


Tuesday, October 24

  • Logic Seminar
    Howard Becker (visiting Caltech)

            A recursion theoretic property of analytic equivalence relations
         3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: Let E be an analytic equivalence relation which does not have perfectly many equivalence classes.  For any oracle a, define L(a,E) to be the set of E-equivalence classes which contain an element y with the property that omega_1^<a,y> = omega_1^a.  For a Turing cone of a's, L(a,E) is countable.



Tuesday, October 31

  • Logic Seminar
    Ronnie Chen (Caltech)

           
       Strong conceptual completeness for L_{\omega_1\omega}
         3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: Strong conceptual completeness (SCC) theorems allow the syntax of a logical theory to be canonically recovered from its space of models equipped with suitable structure, and are known for finitary first-order logic (Makkai) and fragments thereof (Gabriel-Ulmer, Lawvere, and others).  In this talk, I will present a SCC theorem for L_{\omega_1\omega}: a countable L_{\omega_1\omega}-theory can be recovered from its standard Borel groupoid of countable models.


Tuesday, November 7

  • Logic Seminar
    Howard Becker (visiting Caltech)
         Strange structures from computable model theory
        3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: Let L be a countable language, let I be an isomorphism-type of countable L-structures and let a be an oracle.  We say that I is "a-strange" if it contains a recursive-in-a structure and its Scott rank is exactly omega_1^a.  Such structures exist but there are no known natural examples.  Theorem(AD): If C is a collection of aleph_1 isomorphism-types of countable structures, then for a Turing cone of a's, no member of C is a-strange.



Tuesday, November 14

  • Logic Seminar
    Daniel Hoff (UCLA)
         An Operator Algebraic Tool for Deducing Measure Equivalence of Groups
        3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: In this talk, based on joint work with Daniel Drimbe and Adrian Ioana, will focus on an operator algebraic criterion sufficient for deducing measure equivalence of countable groups in the sense of Gromov. In particular, we will give a tool for determining when measure equivalence between $\Gamma_1 \times \Gamma_2$ and $\Lambda_1 \times \Lambda_2$ can be upgraded to measure equivalence between the factors, as is the case in a well known result of Monod and Shalom. The motivation is to bring to bear the power of Sorin Popa's deformation/rigidity theory, but no familiarity with that theory will be assumed.


Tuesday, November 21

  • Logic Seminar
    Michael Hartz (Washington University of St. Louis)
          Interpolating sequences and Kadison-Singer
        3:00 -- 4:00 pm // Mathematics, Room 131


Abstract:
A sequence $(z_n)$ in the unit disc is called an interpolating sequence for $H^\infty$ if for every bounded sequence of complex values $(w_n)$, there exists a bounded analytic function $f$ in the disc such that $f(z_n) = w_n$ for all $n$. Such sequences were characterized by Lennart Carleson.

I will talk about a generalization of Carleson's theorem to other classes of functions. The proof of this result uses the solution of the Kadison-Singer problem due to Marcus, Spielman and Srivastava. This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter.



Tuesday, November 28

  • Logic Seminar
     
    Pieter Spaas (UCSD)

    Non-classification of Cartan subalgebras for a class of von Neumann algebras                                                                 3:00 -- 4:00 pm // Mathematics, Room 131


Abstract: We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We will discuss a construction that leads to a family of II$_1$ factors whose Cartan subalgebras, up to unitary conjugacy, are not classifiable by countable structures. We do this via establishing a strong dichotomy, depending if the action is strongly ergodic or not, on the complexity of the space of homomorphisms from a given equivalence relation to $E_0$. We will start with some of the necessary preliminaries, and then outline the proofs of the aforementioned results.



Tuesday, December 5
  • Logic Seminar
  • Aaron Anderson (Caltech)
    The Fraisse Limit of Finite Dimensional Matrix Algebras with the Rank Metric                                                                           3:00 -- 4:00 pm // Mathematics, Room 131


Abstract: We show that a certain ring, constructed by von Neumann and realized as the coordinatization of a continuous geometry,
can also be realized as the metric Fra ı̈ssé limit of the class of finite-dimensional matrix algebras over a field of scalars,
equipped with the rank metric. We show that the automorphism group of this metric structure is extremely amenable,
implying (by the metric Kechris-Pestov-Todorcevic correspondence) an approximate Ramsey Property, which is also proved
directly.




Monday, January 8

  • Logic Seminar
     
    Clinton Conley (Carnegie Mellon University)

    Unfriendly colorings and path decompositions

       2:00 -- 3:00 pm // Mathematics, Room 131





Wednesday, January 17

  • Logic Seminar
     
    Vibeke Quorning (University of Copenhagen)

    A refined Cantor-Bendixson rank for presented Polish spaces 
           2:00 -- 3:00 pm // Mathematics, Room 105

Abstract: For any Polish space $X$ it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on $F_{\aleph_0}(X)$ if and only if $X$ is a sigma-compact. In the case of $\omega^\omega$ one may recover a co-analytic rank on $F_{\aleph_0}(\omega^\omega)$ by considering the Cantor-Bendixson rank of the induced trees instead. We shall generalize this idea to arbitrarily Polish spaces and thereby construct a family of co-analytic ranks on $F_{\aleph_0}(X)$ for any Polish space $X$. We study the behaviour of this family and compare the obtained ranks to the original Cantor-Bendixson rank. The main results are characterizations of the compact and sigma-compact Polish spaces in terms of this behavior.



Monday, January 22

  • Logic Seminar
    Forte Shinko (Caltech)
    Measure reducibility of countable Borel equivalence relations (after Conley and Miller) 
           2:00 -- 3:00 pm // Mathematics, Room 131

Abstract In a recent Annals paper, Conley and Miller showed that any basis for the countable Borel equivalence relations strictly above E_0 in measure reducibility is uncountable. This is the first in a series of talks where we will provide an overview of this result.








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