Harmonic approximation and its applications

These lectures survey some recent developments concerning the theory and applications of harmonic approximation in Euclidean space. We begin with a discussion of the significance of the concept of thinness for harmonic approximation, and present a complete description of the closed (possibly unbounded) sets on which uniform harmonic approximation is possible. Next we demonstrate the power of such results by describing their use to solve an old problem concerning the Dirichlet problem for unbounded regions. The third lecture characterizes the functions on a given set which can be approximated by harmonic functions on a fixed open superset. Finally, we return to applications, and explain how some problems concerning the boundary behaviour of harmonic functions have recently been solved using harmonic approximation.