Talk 1: Mateusz Olechnowicz - 1:00 pm
Pure Mathematics Department, University of Waterloo
"Introduction to Analytic Continuation"
Let
$f$
be
a
holomorphic
function
defined
around
a
point
$p$
on
a
Riemann
surface
$X$.
Given
any
other
point
$q$
in
$X$,
what
does
it
mean
to
"analytically
continue''
$f$
to
$q$,
and
when
is
it
possible
to
do
so?
The
goal
of
this
talk
is
to
answer
these
questions.
After
reviewing
the
requisite
covering
space
material,
we
will
state
and
prove
the
monodromy
theorem.
We
will
also
discuss
maximal
analytic
continuations,
and
give
an
example
of
a
function
with
no
continuation
beyond
the
disk
within
which
its
defining
power
series
converges.
Talk 2: Ehsaan Hossain - 2:30 pm
Pure Mathematics, University of Waterloo
"Grothendieck's Splitting Theorem"
Recently we have been discussing $\text{Pic}(X)$ and divisors, and we saw that $\text{Pic}(\mathbb{P}^1) \simeq \mathbb{Z}$; more specifically, every line bundle on $\mathbb{P}^1$ is a tensor power of $\mathcal{O}[1]$ or its inverse. It turns out that these constitute all vector bundles over $\mathbb{P}^1$ by taking direct sums. To prove this, we will briefly discuss Serre's Vanishing Theorem and ample line bundles. The proof also uses tools that have been built up throughout the semester.