Richard Gardner

The idea of symmetrization—taking a subset of Euclideanspace (for example) and replacing it by one which preserves some quan-titative aspect of the set but which is symmetric in some sense—is bothprevalent and important in mathematics. The most famous exampleis Steiner symmetrization, introduced by Jakob Steiner around 1838 inhis attempt to prove the isoperimetric inequality (the inequality whichessentially explains why soap bubbles are spheres rather than someother shape). Steiner symmetrization is still a very widely used tool ingeometry, but it and other types of symmetrization are of vital signifi-cance in analysis, PDE’s, and mathematical physics as well.The talk focuses on symmetrization processes that associate to a givenset one that is symmetric with respect to a subspace. In the first phaseof an ongoing joint project with Gabriele Bianchi and Paolo Gronchiof the University of Florence, we consider various properties of an ar-bitrary symmetrization, the relations between these properties, andwhich properties characterize Steiner symmetrization. Several otherwell-known symmetrizations, such as Minkowski symmetrization andcentral symmetrization, will also be discussed. After briefly summa-rizing these results, we shall discuss the second phase, in which weattempt to understand the convergence of iterated symmetrals.