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The present volume grew out of the Heidelberg Knot Theory Semester,
organized by the editors in winter 2008/09 at Heidelberg
University. The contributed papers bring the reader up to date on
the currently most actively pursued areas of mathematical knot
theory and its applications in mathematical physics and cell
biology. Both original research and survey articles are presented;
numerous illustrations support the text. The book will be of great
interest to researchers in topology, geometry, and mathematical
physics, graduate students specializing in knot theory, and cell
biologists interested in the topology of DNA strands.
The present volume grew out of the Heidelberg Knot Theory Semester,
organized by the editors in winter 2008/09 at Heidelberg
University. The contributed papers bring the reader up to date on
the currently most actively pursued areas of mathematical knot
theory and its applications in mathematical physics and cell
biology. Both original research and survey articles are presented;
numerous illustrations support the text. The book will be of great
interest to researchers in topology, geometry, and mathematical
physics, graduate students specializing in knot theory, and cell
biologists interested in the topology of DNA strands.
The central theme of this book is the restoration of Poincare
duality on stratified singular spaces by using Verdier-self-dual
sheaves such as the prototypical intersection chain sheaf on a
complex variety. Highlights include complete and detailed proofs of
decomposition theorems for self-dual sheaves, explanation of
methods for computing twisted characteristic classes and an
introduction to the author's theory of non-Witt spaces and
Lagrangian structures."
Intersection cohomology assigns groups which satisfy a generalized
form of Poincare duality over the rationals to a stratified
singular space. This monograph introduces a method that assigns to
certain classes of stratified spaces cell complexes, called
intersection spaces, whose ordinary rational homology satisfies
generalized Poincare duality. The cornerstone of the method is a
process of spatial homology truncation, whose functoriality
properties are analyzed in detail. The material on truncation is
autonomous and may be of independent interest tohomotopy theorists.
The cohomology of intersection spaces is not isomorphic to
intersection cohomology and possesses algebraic features such as
perversity-internal cup-products and cohomology operations that are
not generally available for intersection cohomology. A
mirror-symmetric interpretation, as well as applications to string
theory concerning massless D-branes arising in type IIB theory
during a Calabi-Yau conifold transition, are discussed.
The central theme of this book is the restoration of Poincare
duality on stratified singular spaces by using Verdier-self-dual
sheaves such as the prototypical intersection chain sheaf on a
complex variety. Highlights include complete and detailed proofs of
decomposition theorems for self-dual sheaves, explanation of
methods for computing twisted characteristic classes and an
introduction to the author's theory of non-Witt spaces and
Lagrangian structures."
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