Jonas Azzam, Universitat Autonoma de Barcelona
"Multiscale
analysis
of
rectifiable
sets"
A
set
E
C
Rn
is
said
to
be
d-rectifiable
if
it
may
be
covered
up
to
d-dimensional
measure
zero
by
Lipschitz
images
of
Rd.
One
can
think
of
these
as
measure-theoretic
analogues
of
differentiable
manifolds:
rather
than
requiring
E
to
be
locally
parametrized
by
smooth
chart-maps,
we
require
E
to
be
parametrized
on
sets
of
positive
measure
by
Lipschitz
maps.
These
are
natural
objects
to
work
with
as
some
results
originally
proven
in
smooth
settings
generalize
to
rectifiable
sets,
and
occasionally
this
is
the
most
general
setting
where
they
can
hold.
They
arise
in
complex
analysis,
the
study
of
the
Dirichlet
problem
in
domains
with
non-smooth
boundaries,
and
the
boundedness
of
singular
integral
operators
on
sets
other
than
Euclidean
space.
In
some
of
these
problems,
knowing
a
set
is
rectifiable
is
not
enough
and
more
quantitative
information
about
the
multiscale
behavior
of
the
set
is
needed.
A
Lipschitz
curve,
for
example,
is
differentiable
almost
everywhere,
so
we
know
that
at
almost
every
point
the
curve
looks
roughly
affine
or
flat
as
we
zoom
in.
A
theorem
of
Peter
Jones,
however,
quantifies
how
often
such
a
curve
is
not
approximately
flat.
In
fact,
he
characterizes
exactly
when
an
arbitrary
set
may
be
contained
in
a
curve
of
finite
length
in
terms
of
a
square
sum
measuring
how
at
the
set
is
at
each
scale
and
location.
This
is
called
the
Analyst's
Traveling
Salesman
Theorem.
In
this
expository
talk,
I
will
give
an
overview
of
the
field
of
quantitative
rectifiability,
its
applications,
and
some
recent
contributions.
DC 1302
Refreshments
will
be
served
in
DC
1302
at
3:30
pm.
Everyone
is
welcome
to
attend.