Author: C.G. Gibson
Publisher: Springer
ISBN: 3540379576
Category : Mathematics
Languages : en
Pages : 160
Book Description
During the academic year 1974-75, the Department of Pure Mathematics in the University of Liverpool held a seminar on the topological stability of smooth mappings. The main objective was to piece together a complete proof of the topological stability theorem (conjectured by René Thom in 1960, and proved by John Mather in 1970) for which no published accounts existed. This volume comprises a write-up of the seminar by four of the participants. Any mathematician working in this area is conscious of a debt to the inventiveness of Thom, and to Mather for the technical work which placed much that was conjecture on firm mathematical foundations. The proof presented in these notes follows Thom's indications closely, and requires no more than some familiarity with differential topology and commutative algebra of the reader.
Topological Stability of Smooth Mappings
Author: C.G. Gibson
Publisher: Springer
ISBN: 3540379576
Category : Mathematics
Languages : en
Pages : 160
Book Description
During the academic year 1974-75, the Department of Pure Mathematics in the University of Liverpool held a seminar on the topological stability of smooth mappings. The main objective was to piece together a complete proof of the topological stability theorem (conjectured by René Thom in 1960, and proved by John Mather in 1970) for which no published accounts existed. This volume comprises a write-up of the seminar by four of the participants. Any mathematician working in this area is conscious of a debt to the inventiveness of Thom, and to Mather for the technical work which placed much that was conjecture on firm mathematical foundations. The proof presented in these notes follows Thom's indications closely, and requires no more than some familiarity with differential topology and commutative algebra of the reader.
Publisher: Springer
ISBN: 3540379576
Category : Mathematics
Languages : en
Pages : 160
Book Description
During the academic year 1974-75, the Department of Pure Mathematics in the University of Liverpool held a seminar on the topological stability of smooth mappings. The main objective was to piece together a complete proof of the topological stability theorem (conjectured by René Thom in 1960, and proved by John Mather in 1970) for which no published accounts existed. This volume comprises a write-up of the seminar by four of the participants. Any mathematician working in this area is conscious of a debt to the inventiveness of Thom, and to Mather for the technical work which placed much that was conjecture on firm mathematical foundations. The proof presented in these notes follows Thom's indications closely, and requires no more than some familiarity with differential topology and commutative algebra of the reader.
Topological Stability of Smooth Mappings
Author: C. G. Gibson
Publisher:
ISBN: 9783662194935
Category :
Languages : en
Pages : 168
Book Description
Publisher:
ISBN: 9783662194935
Category :
Languages : en
Pages : 168
Book Description
Lecture Notes in Mathematics
Author:
Publisher:
ISBN: 9780387079974
Category : Differentiable mappings
Languages : en
Pages : 154
Book Description
Publisher:
ISBN: 9780387079974
Category : Differentiable mappings
Languages : en
Pages : 154
Book Description
Stable Mappings and Their Singularities
Author: M. Golubitsky
Publisher: Springer Science & Business Media
ISBN: 146157904X
Category : Mathematics
Languages : en
Pages : 220
Book Description
This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the theory of singu larities of stable differentiable mappings. The study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R. It was Rene Thorn who noticed (in the late '50's) that all of these results could be incorporated into one theory. The 1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave the first general exposition of this theory. However, these notes preceded the work of Bernard Malgrange [23] on what is now known as the Malgrange Preparation Theorem-which allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subject-and the definitive work of John Mather. More recently, two survey articles have appeared, by Arnold [4] and Wall [53], which have done much to codify the new material; still there is no totally accessible description of this subject for the beginning student. We hope that these notes will partially fill this gap. In writing this manuscript, we have repeatedly cribbed from the sources mentioned above-in particular, the Thom-Levine notes and the six basic papers by Mather.
Publisher: Springer Science & Business Media
ISBN: 146157904X
Category : Mathematics
Languages : en
Pages : 220
Book Description
This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the theory of singu larities of stable differentiable mappings. The study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R. It was Rene Thorn who noticed (in the late '50's) that all of these results could be incorporated into one theory. The 1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave the first general exposition of this theory. However, these notes preceded the work of Bernard Malgrange [23] on what is now known as the Malgrange Preparation Theorem-which allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subject-and the definitive work of John Mather. More recently, two survey articles have appeared, by Arnold [4] and Wall [53], which have done much to codify the new material; still there is no totally accessible description of this subject for the beginning student. We hope that these notes will partially fill this gap. In writing this manuscript, we have repeatedly cribbed from the sources mentioned above-in particular, the Thom-Levine notes and the six basic papers by Mather.
Complements of Discriminants of Smooth Maps
Author: V. A. Vasilʹev
Publisher: American Mathematical Soc.
ISBN: 9780821898376
Category : Mathematics
Languages : en
Pages : 282
Book Description
* Up-to-date reference on this exciting area of mathematics * Discusses the wide range of applications in topology, algebraic geometry, and catastrophe theory.
Publisher: American Mathematical Soc.
ISBN: 9780821898376
Category : Mathematics
Languages : en
Pages : 282
Book Description
* Up-to-date reference on this exciting area of mathematics * Discusses the wide range of applications in topology, algebraic geometry, and catastrophe theory.
Topology of Singular Fibers of Differentiable Maps
Author: Osamu Saeki
Publisher: Springer
ISBN: 3540446486
Category : Mathematics
Languages : en
Pages : 154
Book Description
The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of differential topology. The book contains not only a general theory, but also some explicit examples together with a number of very concrete applications. This is a very interesting subject in differential topology, since it shows a beautiful interplay between the usual theory of singularities of differentiable maps and the geometric topology of manifolds.
Publisher: Springer
ISBN: 3540446486
Category : Mathematics
Languages : en
Pages : 154
Book Description
The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of differential topology. The book contains not only a general theory, but also some explicit examples together with a number of very concrete applications. This is a very interesting subject in differential topology, since it shows a beautiful interplay between the usual theory of singularities of differentiable maps and the geometric topology of manifolds.
Handbook of Geometry and Topology of Singularities III
Author: José Luis Cisneros-Molina
Publisher: Springer Nature
ISBN: 3030957608
Category : Mathematics
Languages : en
Pages : 822
Book Description
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.
Publisher: Springer Nature
ISBN: 3030957608
Category : Mathematics
Languages : en
Pages : 822
Book Description
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.
Singularities of Mappings
Author: David Mond
Publisher: Springer Nature
ISBN: 3030344401
Category : Mathematics
Languages : en
Pages : 567
Book Description
The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings. Part I of the book develops the now classical real C∞ and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic. By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.
Publisher: Springer Nature
ISBN: 3030344401
Category : Mathematics
Languages : en
Pages : 567
Book Description
The first monograph on singularities of mappings for many years, this book provides an introduction to the subject and an account of recent developments concerning the local structure of complex analytic mappings. Part I of the book develops the now classical real C∞ and complex analytic theories jointly. Standard topics such as stability, deformation theory and finite determinacy, are covered in this part. In Part II of the book, the authors focus on the complex case. The treatment is centred around the idea of the "nearby stable object" associated to an unstable map-germ, which includes in particular the images and discriminants of stable perturbations of unstable singularities. This part includes recent research results, bringing the reader up to date on the topic. By focusing on singularities of mappings, rather than spaces, this book provides a necessary addition to the literature. Many examples and exercises, as well as appendices on background material, make it an invaluable guide for graduate students and a key reference for researchers. A number of graduate level courses on singularities of mappings could be based on the material it contains.
Real and Complex Singularities
Author: Laurentiu Paunescu
Publisher: World Scientific
ISBN: 9812705511
Category : Science
Languages : en
Pages : 475
Book Description
The modern theory of singularities provides a unifying theme that runs through fields of mathematics as diverse as homological algebra and Hamiltonian systems. It is also an important point of reference in the development of a large part of contemporary algebra, geometry and analysis. Presented by internationally recognized experts, the collection of articles in this volume yields a significant cross-section of these developments. The wide range of surveys includes an authoritative treatment of the deformation theory of isolated complex singularities by prize-winning researcher K Miyajima. Graduate students and even ambitious undergraduates in mathematics will find many research ideas in this volume and non-experts in mathematics can have an overview of some classic and fundamental results in singularity theory. The explanations are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to go further into the subject and explore the research literature.
Publisher: World Scientific
ISBN: 9812705511
Category : Science
Languages : en
Pages : 475
Book Description
The modern theory of singularities provides a unifying theme that runs through fields of mathematics as diverse as homological algebra and Hamiltonian systems. It is also an important point of reference in the development of a large part of contemporary algebra, geometry and analysis. Presented by internationally recognized experts, the collection of articles in this volume yields a significant cross-section of these developments. The wide range of surveys includes an authoritative treatment of the deformation theory of isolated complex singularities by prize-winning researcher K Miyajima. Graduate students and even ambitious undergraduates in mathematics will find many research ideas in this volume and non-experts in mathematics can have an overview of some classic and fundamental results in singularity theory. The explanations are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to go further into the subject and explore the research literature.
Differential Topology
Author: C. T. C. Wall
Publisher: Cambridge University Press
ISBN: 1316673286
Category : Mathematics
Languages : en
Pages : 355
Book Description
Exploring the full scope of differential topology, this comprehensive account of geometric techniques for studying the topology of smooth manifolds offers a wide perspective on the field. Building up from first principles, concepts of manifolds are introduced, supplemented by thorough appendices giving background on topology and homotopy theory. Deep results are then developed from these foundations through in-depth treatments of the notions of general position and transversality, proper actions of Lie groups, handles (up to the h-cobordism theorem), immersions and embeddings, concluding with the surgery procedure and cobordism theory. Fully illustrated and rigorous in its approach, little prior knowledge is assumed, and yet growing complexity is instilled throughout. This structure gives advanced students and researchers an accessible route into the wide-ranging field of differential topology.
Publisher: Cambridge University Press
ISBN: 1316673286
Category : Mathematics
Languages : en
Pages : 355
Book Description
Exploring the full scope of differential topology, this comprehensive account of geometric techniques for studying the topology of smooth manifolds offers a wide perspective on the field. Building up from first principles, concepts of manifolds are introduced, supplemented by thorough appendices giving background on topology and homotopy theory. Deep results are then developed from these foundations through in-depth treatments of the notions of general position and transversality, proper actions of Lie groups, handles (up to the h-cobordism theorem), immersions and embeddings, concluding with the surgery procedure and cobordism theory. Fully illustrated and rigorous in its approach, little prior knowledge is assumed, and yet growing complexity is instilled throughout. This structure gives advanced students and researchers an accessible route into the wide-ranging field of differential topology.