Handbook of Geometry and Topology of Singularities IV

This is the fourth volume of the Handbook of Geometry and Topology of Singularities, a series that aims to provide 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 which provide an in-depth and reader-friendly survey of various important aspects of singularity theory. Some of these complement topics previously explored in volumes I to III. Amongst the topics studied in this volume are the Nash blow up, the space of arcs in algebraic varieties, determinantal singularities, Lipschitz geometry, indices of vector fields and 1-forms, motivic characteristic classes, the Hilbert-Samuel multiplicity and comparison theorems that spring from the classical De Rham complex.

Singularities are ubiquitous in mathematics and science in general. Singularity theory is a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways.

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

1 Lê Dung Tráng and Bernard Teissier, Limits of tangents, Whitney stratifications and a Plücker type formula.- 2  Anne Frühbis-Krüger and Matthias Zach, Determinantal singularities.- 3 Shihoko Ishii, Singularities, the space of arcs and applications to birational geometry.- 4 Hussein Mourtada, Jet schemes and their applications in singularities, toric resolutions and integer partitions.- 5 Wolfgang Ebeling and Sabir M. Gusein-Zade, Indices of vector fields and 1-forms.- 6 Shoji Yokura, Motivic Hirzebruch class and related topics.- 7 Guillaume Valette, Regular vectors and bi-Lipschitz trivial stratifications in o-minimal structures.- 8 Lev Birbrair and Andrei Gabrielov, Lipschitz Geometry of Real Semialgebraic Surfaces.- 9 Alexandre Fernandes and José Edson Sampaio, Bi-Lipschitz invariance of the multiplicity.- 10 Lorenzo Fantini and Anne Pichon, On Lipschitz Normally Embedded singularities.- 11 Ana Bravo and Santiago Encinas, Hilbert-Samuel multiplicity and finite projections.- 12 Francisco J. Castro-Jiménez, David Mond and Luis Narváez-Macarro, Logarithmic Comparison Theorems.

Lê Dung 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 full-time 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.