Abstract: An embedding $\phi: S \to \mathbb R^4$ from a real surface is called Lagrangian if $\phi^* \omega =0$, where $\omega$ is the standard symplectic form on $\mathbb R^4$. Gromov's theorem (1985) on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of $\mathbb C^n$ provides topological obstructions for Lagrangian embeddings (or immersions) of compact surfaces. However, Givental (1986) showed that such maps always exist if we allow singularities that are either self-intersections or open Whitney umbrellas.

Existence of holomorphic discs attached to a submanifold $X$ of $\mathbb C^n$ is related to the question of polynomial convexity of $X$. I will discuss the joint work with Alexandre Sukhov, where we show that if a Lagrangian surface $X \subset \mathbb C^2$ has an isolated singularity which is a Whitney umbrella, then near the singularity the surface $X$ is locally polynomially convex.