Topological rigidity (and flexibility) of rational and polynomial convexity

Abstract: A compact set in $\mathbb C^n$ is said to be rationally convex if its complement is a union of complex (algebraic) hypersurfaces. A closely related notion is that of a polynomially convex set, which is more simply described as a set cut out by (possibly infinitely many) polynomial inequalities. These definitions have natural analogues in general complex manifolds. Classically, these notions emerged due to approximation-theoretic and Banach-algebraic considerations.  More recently, due to some connections with symplectic geometry, there has been an interest in understanding the topological constraints imposed by such notions of convexity, particularly in two cases: when the compact set is either a closed real submanifold, or the closure of a domain in the ambient space. We will discuss some results of this flavor. 

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