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Seminar 
Thursday,
September 19
2:00  3:00 pm // Linde, Room 255 Abstract: We discuss the interactions
of hyperfiniteness with notions of largeness on Ramsey
spaces. In particular, we present a classical proof of
Mathias's theorem that every countable Borel
equivalence relation on the Ellentuck space is
hyperfinite when restricted to some pure Ellentuck
cube.
This is in joint work with A. Panagiotopoulos
Friday,
October 18
2:00  3:30 pm //UCLA MS 6221 Abstract:
Given a countable
group Gamma, the outer automorphism group
Out(Gamma) is either countable or of cardinality
continuum. A finer and more suitable notion is to
consider the Borel complexity of Out(Gamma) as a
Borel equivalence relation. We show that in this
context, Out(Gamma) is of rather low complexity,
namely that it is a hyperfinite Borel equivalence
relation. In general, we show that for any Polish
group G and any countable normal subgroup Gamma,
the quotient group G/Gamma is hyperfinite. This is
joint work with Joshua Frisch
Tuesday,
October 29
3:00  4:00 pm // Linde, Room 387 Abstract: In
this talk we will go over the proof of the
following theorem of Kathryn Mann: if Homeo(M)
is the group of all homeomorphisms
of a compact manifold M, endowed with the compact
open topology, then every homomorphism from
Homeo(M) to any separable topological group is
necessarily continuous.
Tuesday,
November 5
3:00  4:00 pm // Linde, Room 387 Abstract: In
this talk we will go over the proof of the
following theorem of Kathryn Mann: if Homeo(M)
is the group of all homeomorphisms
of a compact manifold M, endowed with the compact
open topology, then every homomorphism from
Homeo(M) to any separable topological group is
necessarily continuous.
Tuesday,
November 12
3:00  4:00 pm // Linde, Room 387 Abstract: Let Gamma be a countable group. The
invariant random subgroup of a pmp action of Gamma
on X is the measure on the space of subgroups of
Gamma obtained by pushing forward the measure on X
via the map sending x to its stabilizer. A result of
Elek states that if two pmp actions of Gamma have
the same invariant random subgroup and one is
hyperfinite, then they are strongly equivalent, so
in particular they are both hyperfinite. We present
a proof due to Giraud.
Tuesday,
November 26
3:00  4:00 pm // Linde, Room 387 Abstract: It is a well known open problem to
determine if every group is sofic. A sofic group $G$
is said to be flexibly stable if every sofic
approximation to $G$ can converted to a sequence of
disjoint unions of Schreier graphs by modifying an
asymptotically vanishing proportion of edges. We
will discuss a joint result with Lewis Bowen that if
$\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for
some $d \geq 5$ then there exists a group which is
not sofic.
Tuesday,
December 10
2:30  3:30 pm // Linde, Room 387 Abstract: By a flow, we mean a continuous action of a topological groups $G$ on a compact Hausdorff space $X$. We refer to $X$ as the phase space of the flow. We are primarily interested in minimal flows, that is, flows with no nontrivial proper closed invariant subset. Among minimal flows, there exists a maximal one called the universal minimal flow, $M(G),$ which admits a continuous homomorphism onto every minimal flow. When $G$ is nonArchimedean, that is, it admits a neighbourhood basis of the identity of open subgroups, then $M(G)$ is $0$dimensional. These are exactly groups of automorphisms of firstorder structures with the topology of pointwise convergence. If $M(G)$ is $0$dimensional, we can think dually in terms of its algebra of clopen subsets. We summarize which algebras are known to appear as phase spaces of universal minimal flows and we pose questions about the unknown. Friday,
January 10
2:00  3:30 pm //UCLA MS 6221
Abstract: We show that the category of
standard Borel spaces is the free or "universal" category equipped with some familiar set operations of countable arity (e.g., products) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). In this talk, we will discuss the precise formulation of this result, its connection with the amalgamation property for $\kappa$complete Boolean algebras, and its proof using methods from categorical logic. Wednesday,
February 5
2:00  3:00 pm // Linde, Room 387 Abstract: Given a countable group Gamma, there
is a compact space of subgroups Sub(Gamma), which is
equipped with the Gammaaction via conjugation. We
study the notion of weak containment on this space,
namely when one subgroup is contained in the orbit
closure of another, which is related to weak
containment of quasiregular unitary representations
of Gamma. In particular, we will consider necessary
and sufficient conditions for the existence of dense
orbits.
Abstract. A big part of mathematical activity revolves
around classification problems. However, not every
classification problem has a satisfactory solution,
and some classification problems are more
complicated than others. Dynamical properties such
as generic ergodicity and turbulence are crucial in
the development of a rich complexity theory for
classification problems. In this talk we will review
some of the existing anticlassification techniques
and we will introduce a new obstruction for
classification by orbit equivalence relations of TSI
Polish groups; a topological group is TSI if it
admits a compatible two side invariant metric. We
will then show that the Wreath product of any
two noncompact subgroups of $S_{\infty}$ admits an
action whose orbit equivalence relation is
generically ergodic with respect to orbit
equivalence relations of TSI group actions.Wednesday,
February 12
2:00  3:00 pm // Linde, Room 387 Abstract n this talk, we show that any generically E_ Wednesday,
February 19
2:00  3:00 pm // Linde, Room 387 Abstract Wednesday,
April 8
12:00  1:00 pm

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