國立虎尾科技大學 |

Polyfold and Fredholm Theory

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Polyfold and Fredholm Theory/ by Helmut Hofer, Krzysztof Wysocki, Eduard Zehnder.
Author:	Hofer, Helmut.
other author:	Wysocki, Krzysztof.
Description:	XXII, 1001 p.online resource. :
Contained By:	Springer Nature eBook
Subject:	Global analysis (Mathematics). -
Online resource:	https://doi.org/10.1007/978-3-030-78007-4
ISBN:	9783030780074

Polyfold and Fredholm Theory
Hofer, Helmut.

Polyfold and Fredholm Theory[electronic resource] /by Helmut Hofer, Krzysztof Wysocki, Eduard Zehnder. - 1st ed. 2021. - XXII, 1001 p.online resource. - Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,722197-5655 ;. - Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,63.

Part I Basic Theory in M-Polyfolds -- 1 Sc-Calculus -- 2 Retracts -- 3 Basic Sc-Fredholm Theory -- 4 Manifolds and Strong Retracts -- 5 Fredholm Package for M-Polyfolds -- 6 Orientations -- Part II Ep-Groupoids -- 7 Ep-Groupoids -- 8 Bundles and Covering Functors -- 9 Branched Ep+-Subgroupoids -- 10 Equivalences and Localization -- 11 Geometry up to Equivalences -- Part III Fredholm Theory in Ep-Groupoids -- 12 Sc-Fredholm Sections -- 13 Sc+-Multisections -- 14 Extensions of Sc+-Multisections -- 15 Transversality and Invariants -- 16 Polyfolds -- Part IV Fredholm Theory in Groupoidal Categories -- 17 Polyfold Theory for Categories -- 18 Fredholm Theory in Polyfolds -- 19 General Constructions -- A Construction Cheatsheet -- References -- Index.

This book pioneers a nonlinear Fredholm theory in a general class of spaces called polyfolds. The theory generalizes certain aspects of nonlinear analysis and differential geometry, and combines them with a pinch of category theory to incorporate local symmetries. On the differential geometrical side, the book introduces a large class of `smooth’ spaces and bundles which can have locally varying dimensions (finite or infinite-dimensional). These bundles come with an important class of sections, which display properties reminiscent of classical nonlinear Fredholm theory and allow for implicit function theorems. Within this nonlinear analysis framework, a versatile transversality and perturbation theory is developed to also cover equivariant settings. The theory presented in this book was initiated by the authors between 2007-2010, motivated by nonlinear moduli problems in symplectic geometry. Such problems are usually described locally as nonlinear elliptic systems, and they have to be studied up to a notion of isomorphism. This introduces symmetries, since such a system can be isomorphic to itself in different ways. Bubbling-off phenomena are common and have to be completely understood to produce algebraic invariants. This requires a transversality theory for bubbling-off phenomena in the presence of symmetries. Very often, even in concrete applications, geometric perturbations are not general enough to achieve transversality, and abstract perturbations have to be considered. The theory is already being successfully applied to its intended applications in symplectic geometry, and should find applications to many other areas where partial differential equations, geometry and functional analysis meet. Written by its originators, Polyfold and Fredholm Theory is an authoritative and comprehensive treatise of polyfold theory. It will prove invaluable for researchers studying nonlinear elliptic problems arising in geometric contexts.

ISBN: 9783030780074

Standard No.: 10.1007/978-3-030-78007-4doiSubjects--Topical Terms:

1255807
Global analysis (Mathematics).

LC Class. No.: QA614-614.97

Dewey Class. No.: 514.74

Polyfold and Fredholm Theory
LDR:04152nam a22003975i 4500 001 1049458
003 DE-He213
005 20210815004440.0
007 cr nn 008mamaa
008 220103s2021 sz | s |||| 0|eng d
020 $a 9783030780074 $9 978-3-030-78007-4
024 7 $a 10.1007/978-3-030-78007-4 $2 doi
035 $a 978-3-030-78007-4
050 4 $a QA614-614.97
072 7 $a PBKS $2 bicssc
072 7 $a MAT034000 $2 bisacsh
072 7 $a PBKS $2 thema
082 0 4 $a 514.74 $2 23
100 1 $a Hofer, Helmut. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1309461
245 1 0 $a Polyfold and Fredholm Theory $h [electronic resource] / $c by Helmut Hofer, Krzysztof Wysocki, Eduard Zehnder.
250 $a 1st ed. 2021.
264 1 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2021.
300 $a XXII, 1001 p. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, $x 2197-5655 ; $v 72
505 0 $a Part I Basic Theory in M-Polyfolds -- 1 Sc-Calculus -- 2 Retracts -- 3 Basic Sc-Fredholm Theory -- 4 Manifolds and Strong Retracts -- 5 Fredholm Package for M-Polyfolds -- 6 Orientations -- Part II Ep-Groupoids -- 7 Ep-Groupoids -- 8 Bundles and Covering Functors -- 9 Branched Ep+-Subgroupoids -- 10 Equivalences and Localization -- 11 Geometry up to Equivalences -- Part III Fredholm Theory in Ep-Groupoids -- 12 Sc-Fredholm Sections -- 13 Sc+-Multisections -- 14 Extensions of Sc+-Multisections -- 15 Transversality and Invariants -- 16 Polyfolds -- Part IV Fredholm Theory in Groupoidal Categories -- 17 Polyfold Theory for Categories -- 18 Fredholm Theory in Polyfolds -- 19 General Constructions -- A Construction Cheatsheet -- References -- Index.
520 $a This book pioneers a nonlinear Fredholm theory in a general class of spaces called polyfolds. The theory generalizes certain aspects of nonlinear analysis and differential geometry, and combines them with a pinch of category theory to incorporate local symmetries. On the differential geometrical side, the book introduces a large class of `smooth’ spaces and bundles which can have locally varying dimensions (finite or infinite-dimensional). These bundles come with an important class of sections, which display properties reminiscent of classical nonlinear Fredholm theory and allow for implicit function theorems. Within this nonlinear analysis framework, a versatile transversality and perturbation theory is developed to also cover equivariant settings. The theory presented in this book was initiated by the authors between 2007-2010, motivated by nonlinear moduli problems in symplectic geometry. Such problems are usually described locally as nonlinear elliptic systems, and they have to be studied up to a notion of isomorphism. This introduces symmetries, since such a system can be isomorphic to itself in different ways. Bubbling-off phenomena are common and have to be completely understood to produce algebraic invariants. This requires a transversality theory for bubbling-off phenomena in the presence of symmetries. Very often, even in concrete applications, geometric perturbations are not general enough to achieve transversality, and abstract perturbations have to be considered. The theory is already being successfully applied to its intended applications in symplectic geometry, and should find applications to many other areas where partial differential equations, geometry and functional analysis meet. Written by its originators, Polyfold and Fredholm Theory is an authoritative and comprehensive treatise of polyfold theory. It will prove invaluable for researchers studying nonlinear elliptic problems arising in geometric contexts.
650 0 $a Global analysis (Mathematics). $3 1255807
650 0 $a Manifolds (Mathematics). $3 1051266
650 0 $a Functional analysis. $3 527706
650 0 $a Differential geometry. $3 882213
650 0 $a Mathematical analysis. $3 527926
650 0 $a Analysis (Mathematics). $3 1253570
650 1 4 $a Global Analysis and Analysis on Manifolds. $3 672519
650 2 4 $a Functional Analysis. $3 672166
650 2 4 $a Differential Geometry. $3 671118
650 2 4 $a Analysis. $3 669490
700 1 $a Wysocki, Krzysztof. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1353583
700 1 $a Zehnder, Eduard. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1353584
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9783030780067
776 0 8 $i Printed edition: $z 9783030780081
830 0 $a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, $x 0071-1136 ; $v 63 $3 1267908
856 4 0 $u https://doi.org/10.1007/978-3-030-78007-4
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)