Abstract: Given a Calabi-Yau manifold $(M,\omega, J, \Omega)$ of complex dimension $n$, a Special Lagrangian submanifold $L\subset M$ is a real $n$ dimensional submanifold that is a Lagrangian submanifold with respect to the symplectic structure $\omega$ and minimal with respect to the Calabi-Yau metric of the ambient space. In this talk we shall consider singular Special Lagrangian submanifolds in $\mathbb{C}^n$ and their deformation theory. The type of singularities we will consider are the so called edge singularities. These are non-isolated singularities and they are obtained by the process of "edgification" of conical singularities. The deformations of a Special Lagrangian submanifold are given by solutions of a non-linear PDE. We will apply techniques from elliptic PDEs on singular spaces to describe the moduli space of Special Lagrangian deformations with edge singularities. When the obstruction space of this PDE vanishes and by imposing boundary conditions we obtain that the moduli space is a smooth manifold of finite dimension and the deformations obtained are regular solutions of an elliptic boundary value problem (on a singular space) having conormal expansions near the singular set.

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