Handbook of Geometry and Topology of Singularities VII
暫譯: 奇異性幾何與拓撲手冊 第七卷

Cisneros-Molina, José Luis, Dũng Tráng, Lê, Seade, José

Handbook of Geometry and Topology of Singularities VII
  • 出版商: Springer
  • 出版日期: 2025-03-02
  • 售價: $8,980
  • 貴賓價: 9.5 折 $8,531
  • 語言: 英文
  • 頁數: 830
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 3031687108
  • ISBN-13: 9783031687105
  • 相關分類: 離散數學 Discrete-mathematics

    • 海外代購書籍(需單獨結帳)

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商品描述

This is the seventh 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 fourteen chapters that provide an in-depth and reader-friendly introduction to various important aspects of singularity theory. The volume begins with an outstanding exposition on Jim Damon's contributions to singularity theory and its applications. Jim passed away in 2022 and he was one of the greatest mathematicians of recent times, having made remarkable contributions to singularity theory and its applications, mostly to medical image computing.

The next chapter focuses on the singularities of real functions and their bifurcation sets. Then, we look at the perturbation theory of polynomials and linear operators, complex analytic frontal singularities, the global singularity theory of differentiable maps, and the singularities of holomorphic functions from a global point of view.

The volume continues with an overview of new tools in singularity theory that spring from symplectic geometry and Floer-type homology theories. Then, it looks at the derivation of Lie algebras of isolated singularities and the three-dimensional rational isolated complete intersection singularities, as well as recent developments in algebraic K-stability and the stable degeneration conjecture.

This volume also contains an interesting survey on V-filtrations, a theory began by Malgrange and Kashiwara that can be used to study nearby and vanishing cycle functors and introduced by Deligne. Then, we present a panoramic view of the Hodge, toric, and motivic methods in the study of Milnor fibers in singularity theory, both from local and global points of view.

The Monodromy conjecture is also explained; this is a longstanding open problem in singularity theory that lies at the crossroads of number theory, algebra, analysis, geometry, and topology. This volume closes with recent developments in the study of the algebraic complexity of optimization problems in applied algebraic geometry and algebraic statistics.

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

商品描述(中文翻譯)

這是《奇異性幾何與拓撲手冊》第七卷，該系列旨在提供該主題的最新進展、邊界及其與其他研究領域的互動的易懂介紹。

本卷包含十四章，深入且易於閱讀地介紹了奇異性理論的各個重要方面。本卷以對吉姆·達蒙（Jim Damon）對奇異性理論及其應用的貢獻的出色闡述開始。吉姆於2022年去世，他是近代最偉大的數學家之一，對奇異性理論及其應用，尤其是在醫學影像計算方面，做出了卓越的貢獻。

接下來的章節專注於實函數的奇異性及其分岔集。然後，我們將探討多項式和線性算子的擾動理論、複分析前沿奇異性、可微映射的全局奇異性理論，以及從全局觀點看全純函數的奇異性。

本卷繼續概述源自辛幾何和Floer型同調理論的奇異性理論中新工具。接著，探討孤立奇異性的李代數的推導以及三維有理孤立完全交集奇異性，以及代數K穩定性和穩定退化猜想的最新發展。

本卷還包含了一個有趣的調查，關於由Malgrange和Kashiwara提出的V-過濾理論，該理論可用於研究鄰近和消失循環函子，並由Deligne引入。然後，我們從局部和全局的角度呈現Hodge、錐形和動機方法在奇異性理論中研究Milnor纖維的全景視圖。

本卷還解釋了單調性猜想；這是一個在奇異性理論中長期存在的開放問題，位於數論、代數、分析、幾何和拓撲的交叉點。本卷以應用代數幾何和代數統計中優化問題的代數複雜性研究的最新發展作結。

本書適合研究生和初學者，以及可以將其作為指南的專家。

作者簡介

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.

作者簡介(中文翻譯)

何塞·路易斯·西斯內羅斯-莫利納（博士，華威大學 1999）是墨西哥國立自治大學數學研究所的研究員。他的研究興趣包括代數與微分拓撲、微分幾何以及奇異性理論，特別專注於複數和實數解析映射的米爾諾纖維化的一般化。

黎永璋（博士，巴黎大學 1969）是艾克斯-馬賽大學的名譽教授。之前他曾擔任巴黎第七大學（1975-1999）和馬賽大學的教授，並曾擔任三藩市國際理論物理中心（ICTP）數學部門的負責人。他是現代奇異性理論的創始人之一，對於莫爾化、複數奇異性的拓撲、極點變數和旋轉木馬等主題做出了許多貢獻。

何塞·西德（博士，牛津大學 1980）是墨西哥國立自治大學數學研究所的研究員。他的研究集中在奇異變數的向量場指數和陳類的理論，並應用於葉狀結構以及米爾諾的解析映射纖維化定理。他於2021年獲得美洲數學委員會頒發的所羅門·萊夫謝茨獎章。目前他是墨西哥科學院的院長。

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