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This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves. 

While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category. 

The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.

Introduction to Lipschitz Geometry of Singularities: Lecture Notes of the International School on Singularity Theory and Lipschitz Geometry, Cuernavaca, June 2018

 - Geometric Viewpoint of Milnor's Fibration Theorem. - A Quick Trip into Local Singularities of Complex Curves and Surfaces. - 3-Manifolds and Links of Singularities. - Stratifications, Equisingularity and Triangulation. - Basics on Lipschitz Geometry. - Surface Singularities in R4: First Steps Towards Lipschitz Knot Theory. - An Introduction to Lipschitz Geometry of Complex Singularities. - The biLipschitz Geometry of Complex Curves: An Algebraic Approach. - Ultrametrics and Surface Singularities. - Lipschitz Fractions of a Complex Analytic Algebra and Zariski Saturation.