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Logic
Seminar 
Thursday,
October 11
Thursday,
October 25
Abstract: For
a countable Borel equivalence relation E we
consider the weak choice principle ``every
countable sequence of Eclasses 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 Fergodic (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 Fergodic with respect to an
Equasiinvariant measure \mu then the
countable power of E, E^\omega, is Fergodic 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
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.
TuckerDrob, we give a detailed analysis of
the structure of hyperfinite subequivalence
relations of a treed
quasimeasurepreserving 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
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.
TuckerDrob, we give a detailed analysis of
the structure of hyperfinite subequivalence
relations of a treed
quasimeasurepreserving 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.

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