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A unique series of fascinating research papers on subjects
related to the work of Niels Henrik Abel, written by some of the
foremost specialists in their fields. Some of the authors have been
specifically invited to present papers, discussing the influence of
Abel in a mathematical-historical context. Others have submitted
papers presented at the Abel Bicentennial Conference, Oslo June
3-8, 2002. The idea behind the book has been to produce a text
covering a substantial part of the legacy of Abel, as perceived at
the beginning of the 21st century.
This research monograph sets out to study the notion of a local
moduli suite of algebraic objects like e.g. schemes, singularities
or Lie algebras and provides a framework for this. The basic idea
is to work with the action of the kernel of the Kodaira-Spencer
map, on the base space of a versal family. The main results are the
existence, in a general context, of a local moduli suite in the
category of algebraic spaces, and the proof that, generically, this
moduli suite is the quotient of a canonical filtration of the base
space of the versal family by the action of the Kodaira-Spencer
kernel. Applied to the special case of quasihomogenous
hypersurfaces, these ideas provide the framework for the proof of
the existence of a coarse moduli scheme for plane curve
singularities with fixed semigroup and minimal Tjurina number . An
example shows that for arbitrary the corresponding moduli space is
not, in general, a scheme. The book addresses mathematicians
working on problems of moduli, in algebraic or in complex analytic
geometry. It assumes a working knowledge of deformation theory.
Mathematical Models in Science treats General Relativity and
Quantum Mechanics in a non-commutative Algebraic Geometric
framework.Based on ideas first published in Geometry of
Time-Spaces: Non-commutative Algebraic Geometry Applied to Quantum
Theory (World Scientific, 2011), Olav Arnfinn Laudal proposes a Toy
Model as a Theory of Everything, starting with the notion of the
Big Bang in Cosmology, modeled as the non-commutative deformation
of a thick point. From this point, the author shows how to extract
reasonable models for both General Relativity and Quantum Theory.
This book concludes that the universe turns out to be the
6-dimensional Hilbert scheme of pairs of points in affine 3-space.
With this in place, one may develop within the model much of the
physics known to the reader. In particular, this theory is
applicable to the concept of Dark Matter and its effects on our
visual universe.Hence, Mathematical Models in Science proves the
dependency of deformation theory in Mathematical Physics and
summarizes the development of physical applications of pure
mathematics developed in the twentieth century.
This is a monograph about non-commutative algebraic geometry, and
its application to physics. The main mathematical inputs are the
non-commutative deformation theory, moduli theory of
representations of associative algebras, a new non-commutative
theory of phase spaces, and its canonical Dirac derivation. The
book starts with a new definition of time, relative to which the
set of mathematical velocities form a compact set, implying special
and general relativity. With this model in mind, a general Quantum
Theory is developed and shown to fit with the classical theory. In
particular the "toy"-model used as example, contains, as part of
the structure, the classical gauge groups u(1), su(2) and su(3),
and therefore also the theory of spin and quarks, etc.
Noncommutative Deformation Theory is aimed at mathematicians and
physicists studying the local structure of moduli spaces in
algebraic geometry. This book introduces a general theory of
noncommutative deformations, with applications to the study of
moduli spaces of representations of associative algebras and to
quantum theory in physics. An essential part of this theory is the
study of obstructions of liftings of representations using
generalised (matric) Massey products. Suitable for researchers in
algebraic geometry and mathematical physics interested in the
workings of noncommutative algebraic geometry, it may also be
useful for advanced graduate students in these fields.
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