Adjacent Singularities, TQFTs, and Zariski's Multiplicity Conjecture

We give a new proof of Zariski's multiplicity conjecture in the case of isolated hypersurface singularities; this was first proved by de Bobadilla-Pe\l ka \cite{BobadillaPelka}. Our proof uses the TQFT structure of fixed-point Floer cohomology and the fact that adjacent singularities produce symplectic cobordisms between the Milnor fibrations of the singularities. The key technical result is to construct a chain map on Floer cochains using the cobordism and as a last step, apply a spectral sequence of McLean \cite{McLeanLog}. This last step allows us to also recover a theorem of Varchenko \cite{varchenko}.