Alejandra Vicente-Colmenares & Brad Dart, Pure Mathematics Department University of Waterloo
Alejandra Vicente-Colmenares
"Stable co-Higgs bundles"
Abstract:
In
his
thesis
Steven
Rayan
proved
the
non-existence
of
co-Higgs
bundles
over
K3
and
general
type
surfaces,
suggesting
that
some
of
the
interesting
examples
over
surfaces
must
be
found
at
the
lower
end
of
the
Kodaira
classification.
Motivated
by
this,
and
the
fact
that
interesting
co-Higgs
bundles
over
curves
occur
only
over
P1,
studying
co-Higgs
bundles
over
Hirzebruch
surfaces
{starting
with
P1
P1{
seemed
like
a
natural
next
step.
In
this
talk,
the
concept
of
stable
co-Higgs
bundles
will
be
introduced,
and
some
aspects
of
its
classification
over
P1
P1
will
be
discussed.
Brad Dart
"Principal bundles"
Abstract:
Principal
bundles
are
ubiquitous
in
topology
and
differential
geometry,
as
well
as
theoretical
physics;
they
provide
the
mathematical
framework
for
gauge
theories,
for
example,
and
are
a
generalization
of
both
vector
bundles
and
covering
spaces.
A
principal
bundle
consists
of
a
bre
bundle
with
a
free
and
transitive
group
action.
Consequently,
each
bre
is
isomorphic
to
the
group
and
the
base
space
is
isomorphic
to
the
space
of
orbits
under
this
action.
This
definition
and
these
basic
facts
make
sense
in
both
the
topological
and
smooth
categories.
The
frame
bundle
of
a
manifold
is
perhaps
the
most
natural
example
of
a
principal
bundle.
The
possibility
of
further
structure
on
these
bundles,
like
metrics
or
orientations,
is
evidenced
by
a
reduction
of
the
structure
group.
New
bundles
are
generated
via
the
associated
bundle
construction
by
using
representations
of
the
group.
Principal
bundles
can
also
have
a
connection
on
them,
which
is
a
splitting
of
the
tangent
space
into
horizontal
and
vertical
subspaces
that
is
somehow
compatible
with
the
group
action.
This
talk
will
attempt
to
elaborate
and
clarify
as
many
of
these
remarks
as
time
will
allow.