Random reflections, symmetrizations, and foldings on the sphere

Math Physics Seminar
Tuesday, February 17, 2015 - 14:05
1 hour (actually 50 minutes)
Skiles 005
University of Toronto
Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial?  I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).