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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),I
           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.



Wednesday, January 31

  • Logic Seminar
    Forte Shinko (Caltech)
    Measure reducibility of countable Borel equivalence relations (after Conley and Miller), II 
           2:00 -- 3:00 pm // Mathematics, Room 105

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.



Wednesday, February 7

  • Forte Shinko (Caltech)
    Measure reducibility of countable Borel equivalence relations (after Conley and Miller), III 
           3:00 -- 4:00 pm // Mathematics, Room 105

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.



Monday, February 12

  • Logic Seminar
    Forte Shinko (Caltech)
    Measure reducibility of countable Borel equivalence relations (after Conley and Miller), IV
           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.




Wednesday, February 28

  • Logic Seminar
    Joshua Frisch (Caltech)
    Proximal Actions, Strong amenability, and the Infinite Conjugacy Class Property 
           2:00 -- 3:00 pm // Mathematics, Room 105

Abstract: A topological dynamical system (i.e. a group acting by homeomorphisms on a compact  topological space) is said to be proximal if for any two points p and q we can simultaneously push them together i.e. there is a sequence $g_n$ such that $lim g_n(p)=lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $\Z$ acts proximally that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable. In particular I will show a group is strongly amenable if and only if it has no infinite conjugacy class (ICC) groups as factors. This, as a corollary, proves that the only finite generated strongly amenable groups are virtually nilpotent. The proof technique is to show that, for a special class of symbolical systems, a generic (i.e. comeagre) action is proximal. 

This is joint work with Omer Tamuz and Pooya Vahidi Ferdowski.



Tuesday, March 27

  • Logic Seminar
    Asger Tornquist (University of Copenhagen)


             Un-definability of mad families relative to a class of Borel ideals

           1:00 -- 2:00 pm // Mathematics, Room 131

Abstract Many are familiar with the classic result due to Mathias that there are no analytic infinite maximal almost disjoint families,
where almost disjoint means "having finite intersection". In this talk I will discuss what happens if we replace "having finite intersection"
with "having intersection in a (fixed) ideal I", where I is a Borel ideal on $\omega$. It turns out that for a large class of ideals it is
possible to prove an analogue of Mathias' theorem, while for other ideals there _are_ definable infinite mad families. No
characterization of when one or the other situation arises seems to be known at this time.



Monday, April 9

  • Logic Seminar
     
    Ronnie Chen (Caltech)

    Polish groupoids and continuous logic
           1:00 -- 2:00 pm // Mathematics, Room 131

Abstract: It is well-known that every non-Archimedean Polish group is Borel isomorphic to the automorphism group of a countable structure, and analogously, that every Polish group is Borel isomorphic to the automorphism group of a separable metric structure.  We will present a generalization of this result: every open locally Polish groupoid admits a full and faithful Borel functor to the groupoid of metric L-structures on the Urysohn sphere, for some countable metric language L.  This partially answers a question of Lupini.  We will also discuss the analogous result in the non-Archimedean case.


Wednesday, April 25

  • Logic Seminar
    Alekos Kechris (Caltech)
    Co-induction and invariant random subgroups 
           2:00 -- 3:00 pm // Mathematics, Room 131

Abstract: I will discuss a notion of co-induction for invariant random subgroups and show how it can be used to construct new collections of continuum many non-atomic, weakly mixing invariant random subgroups for various classes of groups. I will also discuss an application to the problem of continuity of the co-induction operation on pmp actions. This is joint work with Vibeke Quorning.


Wednesday, May 16

  • Logic Seminar
     
    Tianyi Zheng (UCSD)

      Growth of torsion Grigorchuk groups
           1:00 -- 2:00 pm // Mathematics, Room 131

Abstract: On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. Joint with Anna Erschler.



Tuesday, May 22

  • Logic Seminar
    Jeffrey Bergfalk (Cornell)
  • The cohomology of the first omega alephs: some questions
  •  1:00 -- 2:00 pm // Mathematics, Room 131



Abstract: We describe the conversion of a ladder system on $\omega_n$ to a nonzero element of $\mathrm{H}^n(\omega_n)$ (computed with respect to a constant sheaf on $\omega_n$). In the case of $n=1$, this process of conversion is, simply, Todorcevic's technique of walks on the countable ordinals -- one which, ``[d]espite its simplicity [...] can be used to derive virtually all known other structures that have been defined so far on $\omega_1$'' (\textit{Walks on Ordinals}, 19). For higher $n$, this conversion plays a more prospective role, pointing to dramatically underexplored ZFC structures on higher $\omega_n$. Time permitting, we'll outline two questions in this area that we think of as \textit{next}.


Tuesday, May 29

  • Logic Seminar
     
    Mauro Di Nasso (University of Pisa)
  • Nonstandard natural numbers in Ramsey Theory
           3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: In  Ramsey  Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple points.
In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.





Thursday, May 31

  • Logic Seminar
     
    Peter Burton (University of Texas at Austin)
  • Failure of the L^1 pointwise ergodic theorem for PSL_2(R)
           3:00 -- 4:00 pm // Mathematics, Room 131

Abstract: A 2015 example of Tao showed that the pointwise ergodic theorem for nonabelian free groups can fail for L^1 functions.
 This is in contrast to the situation for L^p, p >1.

We will present joint work with Lewis Bowen showing that this endpoint phenomenon also occurs for PSL_2(R).
The proof involves analyzing the geodesic flow on an appropriately constructed hyperbolic surface.




Tuesday, August 7

  • Logic Seminar
     
    Joseph Zielinski (Carnegie Mellon University)
  • Locally Roelcke precompact Polish groups
           2:00 -- 3:00 pm // Mathematics, Room 131


Abstract: A subset of a Polish group is Roelcke precompact if, given any open subset $ V $ of the group, it can be covered by finitely many sets of the form $ V f V $. Such sets form an ideal and the familiar Roelcke precompact groups are those for which this ideal is improper. A group is said to be locally Roelcke precompact when this ideal countains an open set. Examples of such groups---in addition to all Roelcke precompact groups and all locally compact groups---include the isometry group of the Urysohn metric space and the automorphism group of the countably-regular tree.

All locally Roelcke precompact groups are locally bounded in the sense of the coarse geometry of topological groups developed by C. Rosendal. Indeed, we characterize them as those locally bounded Polish groups for which every coarsely bounded subset is Roelcke precompact. We also characterize them as those groups whose completions with respect to their Roelcke (or lower) uniformities are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup extending multiplication in the group.

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