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Friday, October 27, 2017

**Abstract:** A series of the form $Σa_n z^n, z \in \mathbb{C}$, with radius of convergence $R=1$, is called a Universal Taylor series if for each compact set $K$ which is disjoint from the unit disc and has connected complement, and each function $h$ in $A(K)$, there exists a subsequence of the sequence of partial sums of the series that approximates $h$ uniformly on $K$. In other words, the series diverges so badly outside the unit disc, that it approximates any reasonably good function defined on any reasonably good set. In this talk I will prove the existence of Universal Taylor series and discuss some of their properties.