Abstract: Polynomial convexity is of key importance in the general theory of approximation of continuous functions, uncovering deep connections to topology, Banach algebras, symplectic geometry, and other areas of mathematics. If $S$ is a compact real surface in $\mathbb{C}^2$, a Lagrangian inclusion of S is a map from $S$ to $\mathbb{C}^2$ which is a local Lagrangian embedding, except for a finite number of singularities that are either transverse double points or open Whitney umbrellas. In 1986, Givental proved that any such surface S, orientable or not, admits a Lagrangian inclusion. In this talk we show that Lagrangian inclusions are locally polynomially convex at every point.
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