The principle of topological censorship states that the region outside all black holes in a spacetime should be topologically simple. Classical results show for example that this region is simply connected in 3+1 dimensions.
Recently, an initial data version of topological censorship has been found by Eichmair, Galloway, and Pollack. This states that for a 3-dimensional asymptotically flat initial data set for general relativity the domain outside the outermost marginally trapped surface is diffeomorphic to Euclidean space minus a number of balls.
In this talk I will describe another approach to the initial data version of topological censorship which gives constraints on the topology also in higher dimensions. This is joint work with L. Andersson, G. Galloway, and D. Pollack.

Capillary surfaces are interesting geometric objects which turn out to be important in microgravity environments (e.g. in space) and in tiny devices (e.g. electronic, "lab on a chip"). This talk will focus on the mathematical theory of capillary surfaces in vertical cylinders and sketch the proof of the Concus-Finn conjecture. Related open questions will be mentioned.