The challenge of this project is to produce fundamental advances in the Lipschitz geometry of singularities and, more generally, in the theory of singularities and algebraic geometry, by developing a metric algebraic topology that satisfies the computational tools of ordinary algebraic topology, while also capturing subtle metric phenomena and, in particular, distinguishing objects that are topologically the same but metrically distinct. With this theory in hand, we plan to make significant contributions to several problems in singularity theory and algebraic geometry. Many of these problems are among the most important in their respective fields of study. For example, Zariski’s multiplicity conjecture, which states that the multiplicity of complex hypersurfaces is an invariant of immersed topology, was conjectured more than 50 years ago by Oscar Zariski and is the most important problem in the classical field of equisingularity.