Li Chen
“Symmetry groups of differential equations, Part I.”
Abstract:
The
goal
of
this
talk
is
to
present
a
way
of
discovering
symmetries
hidden
inside
systems
of
differential
equations.
First,
I
will
present
a
survey
of
some
basic
Lie
group
theory.
Then
we
will
consider
actions
of
Lie
groups
on
solutions
of
differential
equations
and
will
pay
special
attention
to
the
ones
that
leaves
the
solutions
invariant.
I
will
also
introduce
the
concept
of
the
prolongation
of
a
Lie
group
action
and
use
it
to
formulate
symmetry
groups
of
differential
equations.
In
the
course
the
of
the
talk,
I
will
also
supply
concrete
examples
of
how
the
theory
works.
This
will
be
the
first
of
two
parts.
Ruxandra
Moraru
will speak on
“Metric connections I: the Levi-Civita connection”
Abstract
This
is
the
first
in
a
series
of
lectures
on
metric
connections.
On
a
smooth
vector
bundle,
a
connection
provides
a
way
of
computing
covariant
derivatives
of
smooth
sections
of
the
bundle
or
of
metrics
on
the
bundle.
If
one
fixes
a
metric
and
the
covariant
derivatives
of
the
metric
vanish
with
respect
to
a
given
connection,
then
this
connection
is
said
to
be
(compatible
with
the)
metric.
In
this
talk,
I
will
begin
by
introducing
connections
on
smooth
vector
bundles
and
present
some
of
their
properties.
I
will
then
specialize
to
affine
connections
on
the
tangent
bundle,
in
particular
giving
a
geometric
interpretation
of
the
curvature
and
the
torsion
of
an
affine
connection.
I
will
finally
discuss
the
Levi-Civita
connection
on
a
Riemannian
manifold,
which
is
the
unique
torsion-free
connection
that
is
compatible
with
the
Riemannian
metric.
Although
I
will
briefly
recall
the
notion
of
vector
bundle,
I
will
assume
the
knowledge
of
differential
forms.