A dynamic adaptive numerical method for solving partial differential
equations (PDEs) on the sphere is necessary to solve problems with
localized structures or sharp transitions. The numerical solution of such
problems on uniform grids is impractical, since high-resolution
computations are required only in regions where sharp transitions occur. An
adaptive wavelet collocation method provides a robust method for
controlling spatial grid adaptation -- fine grid spacing in regions where a
solution varies greatly (i.e., near steep gradients, or near-singularities)
and a much coarser grid where the solution varies slowly, which I will
discuss in this talk. Further, developing a truly adaptive method for
signal analysis is important for the understanding of many natural
phenomena. I will discuss the analysis of a variant of the second order
synchrosqueezing transform to deal with modes containing strong frequency
modulated signal.