Matt DeLand will give the talk on Monday, February 21 at 4:15 in Math 528. This lecture will be open to all.
Abstract: On a compact complex manifold X, it is interesting to ask the question whether or not given a point p, there exists a (non trivial) function holomorphic on X\p with a pole of order v at p. If v = 0, we know that such a function does not exist by Louiville's famous Theorem. If v = 1, we know that such a function does not exist by the Residue Theorem. For higher orders, the answer is sometimes, and it depends on the genus of X and the complex structure of X. We'll ask this question for X a Riemann Surface, and prove the Weierstrass gap theorem as well as some statements as to how many such functions exist, and what the restrictions on them are.
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