Handbook of Geometry and Topology of Singularities III

This is the third volume of the Handbook of Geometry and Topology of Singularities, a series which 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 ten 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 and II, such as, for instance, Zariski's equisingularity, the interplay between isolated complex surface singularities and 3-manifold theory, stratified Morse theory, constructible sheaves, the topology of the non-critical levels of holomorphic functions, and intersection cohomology. Other chapters bring in new subjects, such as the Thom-Mather theory for maps, characteristic classes for singular varieties, mixed Hodge structures, residues in complex analytic varieties, nearby and vanishing cycles, and more.

Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and 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.

這是《奇異點的幾何和拓撲手冊》的第三卷，這個系列旨在提供對該主題的最新狀態、前沿和與其他研究領域的交互作用的易於理解的描述。本卷包含十個章節，深入且易於閱讀地介紹了奇異點理論的各個重要方面。其中一些章節補充了之前在第一卷和第二卷中探討的主題，例如Zariski的等奇性、孤立複雜曲面奇異點與三維流形理論之間的相互作用、分層莫爾斯理論、可構造層、全純函數的非臨界層的拓撲以及交叉上同調。其他章節引入了新的主題，例如Thom-Mather理論、奇異變量的特徵類、混合霍奇結構、複解析變量中的殘差、附近和消失週期等。

奇異點在數學和科學中無處不在。奇異點理論與其他數學領域有著活躍的交互作用，作為一個容器，不同類型的數學問題相互作用，令人驚訝的聯繫產生，簡單的問題引發的思想在主題的其他部分以及其他學科中產生共鳴。由世界專家撰寫，各種貢獻涵蓋了經典材料和現代發展，涉及了各種基本相互關聯的主題。

本書針對研究生和新手，以及專家們作為指南。

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

Dũng Tráng Lê (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, 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. In 2007 he was awarded the Ferran Sunyer i Ballaguer prize, and he was awarded the 2021 Solomon Lefschetz Medal by the Mathematical Council of the Americas.

José Luis Cisneros-Molina（1999年華威大學博士）是墨西哥國立自治大學數學研究所的全職研究員。他的研究興趣包括代數和微分拓撲、微分幾何和奇異性理論，尤其關注複解析和實解析映射的Milnor纖維的推廣。

Dũng Tráng Lê（1969年巴黎大學博士）是Aix-Marseille大學的名譽教授。此前，他曾任巴黎第七大學（1975-1999年）和馬賽大學的教授，並擔任過國際理論物理中心（ICTP）的數學主任。作為現代奇異性理論的創始人之一，他在莫爾斯化、複奇異性的拓撲、極性變量、旋轉木馬等領域做出了眾多貢獻。

José Seade（1980年牛津大學DPhil）是墨西哥國立自治大學數學研究所的全職研究員。他的研究領域包括向量場的指數理論和奇異變量的Chern類理論，並應用於葉狀結構和Milnor的解析映射纖維定理。他於2007年獲得Ferran Sunyer i Ballaguer獎，並於2021年獲得美洲數學理事會頒發的Solomon Lefschetz獎章。