# Vector bundles with sections

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Brian Osserman, University of California, Davis
Fine Hall 322

Classical Brill-Noether theory studies, for given $g, r, d$, the space of line bundles of degree $d$ with $r+1$ global sections on a curve of genus $g$. We will review the main results in this theory, and the role of degeneration techniques in proving them, and then we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open.