Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.