Pure Mathematics Department, University of Waterloo
Matthew Beckett “Non-minimal Yang-Mills solutions”
Abstract:
Connections
which
satisfy
the
Yang-Mills
equations
are
critical
points
of
the
Yang-Mills
action.
Minimal
points
on
the
4-sphere
are
well
understood
via
the
ADHM
construction,
but
it
is
a
natural
question
to
ask
whether
there
exist
non-minimal
critical
points.
In
this
talk
I
will
present
some
results
from
a
1991
paper
by
Thomas
H.
Parker
which
constructs
some
such
examples
of
non-minimal
Yang-Mills
solutions.
Robert Garbary “Elliptic curves and the group law”
Abstract:
In
number
theory,
an
elliptic
curve
is
the
zero
locus
of
a
smooth
irreducible
cubic
polynomial
in
P2.
Such
an
object
comes
equipped
with
a
group
law,
and
much
of
number
theory
is
concerned
with
studying
this
group
law.
However,
the
group
law
has
a
very
strange
definition
—
you
add
two
points
by
doing
two
different
intersections
of
lines
with
the
curves.
In
this
talk,
I
will
show
that
the
group
law
is
a
consequence
of
the
Riemann–Roch
theorem.
In
particular,
for
a
compact
Riemann
surface
X
of
genus
1,
it
implies
a
(non-canonical)
bijection
between
X
and
Pic0(X)
—
the
group
of
degree
zero
line
bundles
on
X.
Using
this
bijection,
we
can
define
a
group
law
on
X.
I
will
show
that,
in
the
case
that
X
is
embedded
into
P2
as
a
smooth
cubic
(every
X
admits
such
an
embedding),
this
group
law
is
exactly
the
one
that
people
in
number
theory
care
about.