PIMS - UBC Distinguished Colloquium: Laurent Saloff-Coste

The "doubling property" refers to the property (of a metric measure space) that max{Vol(B(x,2r))/Vol(B(x,r)): r>0} is bounded.  We consider the following question:  do we have good control of the doubling property for left-invariant geometries on a given compact Lie group? For instance, on the group SU(2) (which, as a manifold, is the 3-sphere) what can we say of the doubling constant of a left-invariant geometry?   We will discuss the conjecture that, for any compact Lie group G, there is a constant D(G) such that max{Vol_g(2r)/Vol_g(r): r>0} is bounded by D(G) uniformly over all left-invariant metric g. This is true in the case of SU(2).  This talk is based on joint work with Nate Eldredge and Maria Gordina.