This is the eight volume of the Handbook of Geometry and Topology of Singularities, a series that provides an accessible account of the state of the art of the subject, its frontiers and its interactions with other areas of research. 

This volume consists of twelve chapters with reader-friendly introductions to several important topics and aspects of singularity theory, such as:

• Plane curve singularities studied by means of divides, which capture a lot of their topology.
• Viro’s method to study the topology of real algebraic varieties, providing a wide range of  possible combinations of topological and combinatorial invariants.
• Local tropicalization, a technique for attaching a combinatorial object to germs of subvarieties of algebraic tori and toric varieties.
• The theory of Zariski pairs and superisolated singularities.
• The McKay correspondence, a deep connection that links group theory, algebraic geometry, and representation theory.
• Calculations with Characteristic Cycles, a deep concept in the interplay between algebraic geometry, representation theory and microlocal analysis.
• The monodromy zeta functions in singularity theory.
• The singularities of the minimal model program of complex quasi-projective varieties.
• A general theory of Thom polynomials associated to the classification of map-germs.
• A discussion on indices and residues, intertwining the theories of complex analytic singular varieties and singular holomorphic foliations.
• The Monodromy in Integral Geometry and PDE.
• The topological theory of Hyperplane Arrangements.

 The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

José Luis Cisneros-Molina (PhD, University of Warwick 1999) is a researcher at the Mathematics Institute of the National Autonomous University of Mexico. His research interests include Algebraic and Differential Topology, Differential Geometry, and Singularity Theory, with a particular focus on generalizations of Milnor Fibrations for complex and real analytic maps.

Lê Dũng Tráng (PhD, University of Paris 1969) is an Emeritus Professor at Aix-Marseille University. Previously he was Professor at the Universities of Paris VII (1975-1999) and Marseille, and was head of Mathematics at the ICTP at Trieste. One of the founders of modern Singularity Theory, he has made numerous contributions to morsification, the topology of complex singularities, polar varieties, and carousels, among other topics.

José Seade (DPhil, University of Oxford 1980) is a researcher at the Mathematics Institute of the National Autonomous University of Mexico. His research is in the theory of indices of vector fields and Chern classes for singular varieties, with applications to foliations, and Milnor’s Fibration theorem for analytic maps. He was awarded the 2021 Solomon Lefschetz Medal by the Mathematical Council of the Americas. He is currently President of the Mexican Academy of Sciences.