We will begin this talk with a brief summary of different notions of equisingularity in families of isolated hypersurface singlarities, concentrating on various notions of simultaneous deisngularization.
We will then report on joint work in progress with Max Leyton and Hussein Mourtada. The starting point of this work is a 1980 paper by Yujiro Kawamata in which the author claims to relate the existence of simultaneous embedded resolution in a family to the property of it being µ*-constant. Inspired by this paper, we formulated two conjectures relating the notions of µ-constant, µ*-constant and simultaneous embedded resolution. We will illustrate the first conjecture with the example of Birançon-Speder.