Abstract: This week we are covering: ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

Abstract: Let $E$ be a closed subset in the complex plane with connected complement. We define $A(E)$ to be the class of all complex continuous functions on $E$ that are holomorphic in the interior $E^0$ of $E$. The remarkable theorem of Mergelyan shows that every $f\in A(E)$ is uniformly approximable by polynomials on $E$, but is it possible to realize such an approximation by polynomials that are zero-free on $E$? This question was first proposed by J.Anderson and P.Gauthier. Recently Arthur Danielyan described a class of functions for which zero-free approximation is possible on an arbitrary $E$. I am intending to talk about the generalization of his work on Riemann surfaces.

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