Mathematical Conversations

The harmonic measure is an important tool, which allows one to reconstruct a harmonic function from its values on the boundary. But it also admits a very simple and beautiful probabilistic interpretation: it is the probability that the path of the Brownian motion originating at a specified point will exit the domain through the given portion of the boundary, in other words, it shows how well a Brownian traveler "sees" certain pieces of the boundary. The relations between the properties of this measure and geometry of the domain and its boundary are deep and mysterious: despite a century of research some questions are still open and some have only been resolved recently. We shall discuss some known connections as well as unsolved problems.