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We will start by defining polynomial convexity of a
compact subset of $\mathbb{C}^n$ and discussing its meaning in
dimension one. Next, we will discuss the connection between
polynomial convexity and uniform approximations by holomorphic
polynomials. In general, given a compact subset of $\mathbb{C}^n$,
$n\geq 2$, it is very difficult to determine whether it is
polynomially convex. In this talk we will present a necessary and
sufficient condition for polynomial convexity of a compact subset
when the given compact subset lies in a totally-real submanifold of
$\mathbb{C}^n$, $n\geq 2$.
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