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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Functional analysis
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Polyfold and Fredholm Theory (Hardcover, 1st ed. 2021)
Loot Price: R6,514
Discovery Miles 65 140
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Polyfold and Fredholm Theory (Hardcover, 1st ed. 2021)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 72
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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.
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