Kate Juschenko, Northwestern University
"Techniques
and
concepts
of
amenability"
The
subject
of
amenability
essentially
begins
in
1900's
with
Lebesgue.
He
asked
whether
the
properties
of
his
integral
are
really
fundamental
and
follow
from
more
familiar
integral
axioms.
This
led
to
the
study
of
positive,
finitely
additive
and
translation
invariant
measure
on
reals
as
well
as
on
other
spaces.
In
particular
the
study
of
isometry-invariant
measure
led
to
the
Banach-Tarski
decomposition
theorem
in
1924.
The
class
of
amenable
groups
was
introduced
by
von
Neumann
in
1929,
who
explained
why
the
paradox
appeared
only
in
dimensions
greater
or
equal
to
three,
and
does
not
happen
when
we
would
like
to
decompose
the
two-dimensional
ball.
In
1940's,
M.
Day
defined
a
class
of
elementary
amenable
groups
as
the
largest
class
of
groups
amenability
of
which
was
known
to
von
Naumann.
He
asked
whether
there
are
other
groups
then
that.
We
will
give
an
introductory
to
amenability
talk,
and
explain
more
recent
developments
in
this
eld.
In
particular,
I
will
explain
how
to
obtain
all
known
non-elementary
amenable
groups
using
only
one
approach.
MC
5501
Refreshments
will
be
served
in
MC
5501
at
3:30
pm.
Everyone
is
welcome
to
attend.