Friday, November 22, 2019 4:00 pm
-
4:00 pm
EST (GMT -05:00)
Kiumars Kaveh, University of Pittsburg,
"Valuations
in
algebraic
geometry
and
applications"
We
start
by
an
overview
of
the
basic
notion
of
a
valuation
(on
an
algebra)
and
construction
of
convex
polytopes
and
convex
bodies
in
algebraic
geometry
via
valuations.
This
is
in
the
heart
of
general
theory
of
Newton-Okounkov
bodies.
The
main
theorem
is
an
elegant
formula
for
the
number
of
intersection
points
of
hypersurfaces
in
a
projective
variety
in
terms
of
volume
of
convex
bodies.
In
symplectic
geometry
this
gives
strong
results
about
Hamiltonian
torus
actions
on
projective
varieties
and
constructing
symplectic
ball
embeddings.
The
above
concerns
study
of
"line
bundles"
on
varieties.
We
then
consider
extending
these
ideas
to
more
general
bundles
such
as
"vector
bundles"
and
"principal
bundles"
and
state
our
very
recent
results
on
classifying
toric
principal
bundles.
This
is
a
fruitful
interaction
of
ideas
from
algebraic
geometry,
building
theory
and
tropical
geometry.
For
the
most
part
the
talk
should
be
accessible
to
anybody
with
basic
knowledge
of
algebra
and
geometry.
MC
5479
Refreshments
will
be
served
at
5:00pm
in
MC
5403