**CANCELLED**
Ragnar-Olaf Buchweitz, University of Toronto
"The McKay Correspondence Then and Now"
In
his
treatise
on
"Symmetry",
Hermann
Weyl
credits
Leonardo
Da
Vinci
with
the
insight that
the
only
finite
symmetry
groups
in
the
plane
are
cyclic
or
dihedral.
Reaching
back
even
farther,
had
the
abstract
notion
been
around,
Euclid's
Elements may
well
have
ended
with
the
theorem
that
only
three
further
groups
can
occur
as finite
groups
of
rotational
symmetries
in
3-space,
namely
those
of
the
Platonic
solids. Of
course,
it
took
another
22
centuries
for
such
a
formulation
to
be
possible,
put
forward
by C. Jordan
(1877)
and
F.
Klein
(1884).
Especially
Klein's
investigation
of
the
orbit
spaces
of
those
groups
and
their
double
covers, the
binary
polyhedral
groups,
is
at
the
origin
of
singularity
theory
and
in
the
century
afterwards many
surprising
connections
with
other
areas
of
mathematics
such
as
the
theory
of
simple
Lie
groups were
revealed
in
work
by
Grothendieck,
Brieskorn,
and
Slodowy
in
the
1960's
and
70's. A
beautiful
and
comprehensive
survey
of
that
side
of
the
story
was
given
by
G.-M.
Greuel in
the
extended
published
version
of
his
talk
at
the
Centennial
Meeting
of
the
DMV
in
1990 in
Bremen.
It
came
then
as
a
complete
surprise
when
J.
McKay
pointed
out
in
1979
a
very
direct,
though
then mysterious
relationship
between
the
geometry
of
the
resolution
of
singularities
of
these
orbit
spaces and
the
representation
theory
of
the
finite
groups
one
starts
from.
In
particular,
he
found
a remarkably
simple
explanation
for
the
occurrence
of
the
Coxeter-Dynkin
diagrams
in
the
theory.
This
marks
essentially
the
beginning
of
"Noncommutative
Singularity
Theory",
the
use
of
representation theory
of
not
necessarily
commutative
algebras
to
understand
the
geometry
of
singularities,
a
subject
area that
has
exploded
during
the
last
decade
in
particular
because
of
its
role
in
the
mathematical formulation
of
String
Theory
in
Physics.
In
this
talk
I
will
survey
the
beautiful
classical
mathematics
at
the
origin
of
this
story and
then
give
a
sampling
of
recent
results
and
of
work
still
to
be
done.
MC 5501