Real Analytic And Algebraic Singularities


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Real Analytic and Algebraic Singularities


Real Analytic and Algebraic Singularities

Author: Toshisumi Fukui

language: en

Publisher: CRC Press

Release Date: 1997-12-12


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This book contains a collection of papers covering recent progress in a number of areas of singularity theory. Topics include blow analyticity, recent progress in the research on equivalence relations of maps and functions, sufficiency of jets, and the transversality theorem. . Geometric and analytic studies of partial differential equations have been developed independently of one another, but the shock wave solutions appearing in natural phenomena are not well understood. Singularity theory may unify these studies and a survey based on this viewpoint is presented in which a new notion of weak solution is introduced. There are also reports on the recent progress in Zariski's conjecture on multiplicities of hypersurfaces, transcendency of analytic sets and on the topology of weighted homogeneous polynomials. This book will be of particular interest to specialists in singularities, partial differential equations, algebraic geometry and control theory.

Real and Complex Singularities


Real and Complex Singularities

Author: Laurentiu Paunescu

language: en

Publisher: World Scientific

Release Date: 2007


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The modern theory of singularities provides a unifying theme that runs through fields of mathematics as diverse as homological algebra and Hamiltonian systems. It is also an important point of reference in the development of a large part of contemporary algebra, geometry and analysis. Presented by internationally recognized experts, the collection of articles in this volume yields a significant cross-section of these developments. The wide range of surveys includes an authoritative treatment of the deformation theory of isolated complex singularities by prize-winning researcher K Miyajima. Graduate students and even ambitious undergraduates in mathematics will find many research ideas in this volume and non-experts in mathematics can have an overview of some classic and fundamental results in singularity theory. The explanations are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to go further into the subject and explore the research literature.

Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry


Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry

Author: Michel Coste

language: en

Publisher:

Release Date: 2007


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In this volume the authors present some new trends in real algebraic geometry based on the study of arc spaces and additive invariants of real algebraic sets. Generally, real algebraic geometry uses methods of its own that usually differ sharply from the more widely known methods of complex algebraic geometry. This feature is particularly apparent when studying the basic topological and geometric properties of real algebraic sets; the rich algebraic structures are usually hidden and cannot be recovered from the topology. The use of arc spaces and additive invariants partially obviates this disadvantage. Moreover, these methods are often parallel to the basic approaches of complex algebraic geometry. The authors' presentation contains the construction of local topological invariants of real algebraic sets by means of algebraically constructible functions. This technique is extended to the wider family of arc-symmetric semialgebraic sets. Moreover, the latter family defines a natural topology that fills a gap between the Zariski topology and the euclidean topology. In real equisingularity theory, Kuo's blow-analytic equivalence of real analytic function germs provides an equivalence relation that corresponds to topological equivalence in the complex analytic set-up. Among other applications, arc-symmetric geometry, via the motivic integration approach, gives new invariants of this equivalence, allowing some initial classification results. The volume contains two courses and two survey articles that are designed for a wide audience, in particular students and young researchers.