Title: The Shapiro-Shapiro Conjecture
| Speaker: | Kevin Purbhoo |
| Affiliation: | University of Waterloo |
| Room: | MC 5501 |
Abstract:
Given
four
lines
in
3-space,
can
you
find
a
fifth
line
that
intersects the
other
four? How
many?
This
is
the
smallest
non-trivial
example
of
a
"Schubert
problem".
The
answer,
in
this
case,
is
not
hard
to
compute:
there
are
two
such
lines.
Generalizations
of
this
fact
date
back
to
19th
century
work
of
Schubert.
However,
this
answer
comes
with
the
usual
caveats
of
enumerative
geometry:
not
always,
not
necessarily
over
the
reals,
and
what
exactly
do
you mean
by
"find"?
Addressing
these
finer
points
is
much
harder,
and
until
fairly
recently
not
much
was
known.
In
1993,
Boris
and
Michael
Shapiro
formulated
a
remarkable
conjecture
about
the
existence
of
real
solutions
to
Schubert
problems. It
was
proved
in
2009 by
Mukhin,
Tarasov
and
Varchenko,
using
high
powered
machinery
from
quantum integrable
systems. Since
then,
many
applications
and
generalizations
have
been
found
or
conjectured.
In
this
talk,
I
will
tell
the
story
of
the
Shapiro-Shapiro
conjecture,
and
then
tell
you
about
a
new
theorem
(joint
work
with
Jake
Levinson),
a not-so-obvious
generalization. Our
result
is
independent
of Mukhin-Tarasov-Varchenko,
implies
the
conjecture,
and
naturally
lends itself
to
a
much
more
intuitive
proof.