Address: Mathematics
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Seminar 
Tuesday,
October 3
Tuesday,
October 17
Tuesday,
October 24
A
recursion theoretic property of
analytic equivalence relations
Tuesday,
October 31
Tuesday,
November 7
3:00  4:00 pm // Mathematics, Room 131 Abstract: Let L be a countable language, let I be an isomorphismtype of countable Lstructures and let a be an oracle. We say that I is "astrange" if it contains a recursiveina 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 isomorphismtypes of countable structures, then for a Turing cone of a's, no member of C is astrange. Tuesday,
November 14
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
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 KadisonSinger problem due to Marcus, Spielman and Srivastava. This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter. Tuesday,
November 28
Tuesday, December 5
Monday,
January 8
2:00  3:00 pm // Mathematics, Room 131
Wednesday,
January 17
Abstract: For any Polish space $X$ it is wellknown that the CantorBendixson rank provides a coanalytic rank on $F_{\aleph_0}(X)$ if and only if $X$ is a sigmacompact. In the case of $\omega^\omega$ one may recover a coanalytic rank on $F_{\aleph_0}(\omega^\omega)$ by considering the CantorBendixson rank of the induced trees instead. We shall generalize this idea to arbitrarily Polish spaces and thereby construct a family of coanalytic 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 CantorBendixson rank. The main results are characterizations of the compact and sigmacompact Polish spaces in terms of this behavior. Monday,
January 22
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
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
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
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
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 nonamenable 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
Undefinability of mad families relative to a class of Borel ideals 1:00  2:00 pm // Mathematics, Room 131Abstract: 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
Abstract: It is wellknown that every nonArchimedean 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 Lstructures 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 nonArchimedean case. Wednesday,
April 25
Abstract: I will discuss a notion of coinduction for invariant random subgroups and show how it can be used to construct new collections of continuum many nonatomic, weakly mixing invariant random subgroups for various classes of groups. I will also discuss an application to the problem of continuity of the coinduction operation on pmp actions. This is joint work with Vibeke Quorning. Wednesday,
May 16
Abstract: On torsion Grigorchuk groups we construct random walks of finite entropy and powerlaw tail decay with nontrivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. Joint with Anna Erschler. Tuesday,
May 22
Tuesday,
May 29
Abstract: In Ramsey Theory, ultrafilters often play an instrumental role. By using nonstandard models of the integers, one can replace those thirdorder 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
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
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 groupsin addition to all Roelcke
precompact groups and all locally compact
groupsinclude the isometry group of the Urysohn
metric space and the automorphism group of the
countablyregular 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 semitopological
semigroup extending multiplication in the group.

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