Waterloo Geometry Summer School 2016
Register
Title:
On
the
volume
spectrum
in
hyperbolic
geometry
Speaker: Luca Di Cerbo, International Centre for Theoretical Physics (ICTP) (Mathematics Section), Trieste, Italy
Lecture
1:
Wednesday
June
29,
1:30
p.m.
-
3:00
p.m.,
MC
5501
The
study
of
volumes
in
hyperbolic
geometry,
real
&
complex,
is
a
classical
topic
in
differential
geometry
which
is
beautifully
interconnected
with
many
other
parts
of
mathematics
such
as
geometric
topology,
algebraic
geometry,
group
theory
and
even
computational
mathematics.
In
this
first
lecture,
I
will
introduce
this
fascinating
subject
through
the
study
of
volumes
of
hyperbolic
Riemann
surfaces
via
the
classical
Gauss-Bonnet
theorem.
Then,
I
will
give
an
overview
of
some
modern
works
on
hyperbolic
3-manifolds
due
to
Thurston
and
his
school.
Finally,
I
will
recall
the
necessary
background
for
the
study
of
volumes
in
higher
dimensional
complex
hyperbolic
geometry
such
as
the
Chern-Gauss-Bonnet
theorem.
The
last
part
of
this
lecture
is
preparatory
for
the
second
lecture,
which
will
be
exclusively
on
complex
hyperbolic
geometry.
Lecture
2:
Thursday
June
30,
1:30
p.m.
-
3:00
p.m.,
MC
5501
In
this
lecture,
I
will
focus
on
the
study
of
complex
hyperbolic
surfaces
and
their
volume
spectrum.
Recall
that
a
complex
hyperbolic
surface
is
a
complete
finite
volume
complex
surface
(a
manifold
of
real
dimension
four)
which
admits
a
metric
of
negative
constant
holomorphic
sectional
curvature.
In
the
compact
case,
a
problem
which
has
attracted
considerable
attention
is
the
classification
of
all
complex
hyperbolic
surfaces
with
minimal
volume.
This
intriguing
problem,
originally
proposed
by
Mumford
in
the
70's,
has
a
long
and
rich
history
which
I
will
briefly
recall.
The
analogous
problem
in
the
non-compact
finite
volume
case
has
an
interesting
history
as
well.
In
fact,
Hirzebruch
in
the
80's
constructed
the
first
example
of
a
non-compact
complex
hyperbolic
surface
with
minimal
volume.
After
reviewing
Hirzebruch's
example,
I
will
construct
a
new
example,
recently
discovered
in
collaboration
with
M.
Stover,
and
address
the
classification
problem
of
all
such
surfaces.
Finally,
I
will
explicitly
construct
a
tower
of
examples
which
saturates
the
whole
admissible
volume
spectrum
of
complex
hyperbolic
surfaces.
This
is
again
joint
with
M.
Stover.
Registration for this event is required.
For more information contact, Benoit Charbonneau.
To help us advertise, please download a copy of the event poster.