Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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Pure Mathematics Department, University of Waterloo
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.
Pure Mathematics, University of Waterloo
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.
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca