Harmonic maps, inverse problems, and related topics

In this thesis we consider Calderón's problem for harmonic maps in real-analytic setting. In the first chapter we provide foundational and background material such as the existence and uniqueness of a solution to the Dirichlet problem for the connection Laplacian, and the existence and uniqueness of the Dirichlet Green kernel. In the second chapter we discuss the properties of the Dirichlet-to-Neumann operator associated to the connection Laplacian and prove a result on reconstruction of geometric data on the boundary from a given Dirichlet-to-Neumann operator. We then use this to prove a uniqueness result for Calderón's inverse problem for the connection Laplacian on a vector bundle. In the third chapter we generalise the notion of the Dirichlet-to-Neumann operator to maps between manifolds and discuss what kind of difficulties arise along the way. We conclude the chapter with the uniqueness result for Calderón's inverse problem for maps between real-analytic manifolds.