Equimultiplicity of mu-constant families

I will present my recent joint work with J. F. de Bobadilla, proving that a family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. The key idea is to endow the A'Campo model of "radius zero" monodromy with a symplectic structure. More precisely, I will construct a smooth manifold, fibered over an annulus, together with a fiberwise symplectic form such that the following holds. Over the outer circle ("positive radius"), we get the usual Milnor fibration. In turn, over the inner circle ("radius zero"), the symplectic monodromy has very simple dynamical properties. Thus we can compute its fixed point Floer homology using a generalized version of the McLean's spectral sequence, which recovers the multiplicity. If time permits, I will outline possible applications of such a "radius zero" construction to other degeneration problems.