Abstract: We consider invariant point processes, i.e. random collections of points with distribution invariant under isometries: the simplest example is the Poisson point process. Given a point process M in the plane, the Voronoi tesselation assigns a polygon (of different area) to each point of M.

The geometry of "fair" allocations is much richer: There is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem.

Zeros of power series with Gaussian coefficients are a different source of point processes, where the isometry invariance is connected to classical complex analysis. In the case of independent coefficients with equal variance, the zeros form a determinantal process in the hyperbolic plane, with conformally invariant dynamics. Surprisingly, in this case the number of zeros in a disk has a coin-tossing interpretation.

The talk is based on joint works with C. Hoffman, A. Holroyd and B. Virag.