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.