Aspects of Mathematics, Volume: 39
2010; 181 pp; hardcover
ISBN-13: 978-3-8348-1271-1
The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics.
This volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras.
The papers originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn.
Graduate students and research mathematicians interested in algebraic geometry.
Contributions by Gregory Ginot, Thomas M. Fiore and Igor Kriz, Toshiro Hiranouchi and Satoshi Mochizuki, Paulo Carrillo Rouse, Donatella Iacono and Marco Manetti, John Terilla, and Anne Pichereau.
The book is a compilation of selected papers from the conference on Physics and Control 2009, presenting a unified perspective underlying the thematics and strategies related to the control of physical systems with emerging applications in physics, engineering, chemistry, biology and other natural sciences. The selected papers reflect the state-of-the-art of the more advanced theoretical and practical studies in the field of control of complex systems. The contributions provide a comprehensive view on some selected topics of particular importance at the disciplinary borderline between Physics and Control.
Modelling and Control of Coupled Stochastic Oscillators
Multistability in Natural and Laboratory-Scale Nonlinear Systems
Linear and Matritial Algebra, Open Problems Related to Control Theory
Localization of Oscillations in Dynamical Systems
Microfluidics: Theory, Methods and Applications
Mathematical Modelling of Dynamic Systems for Volcano Physics
Geometric Control for Quantum and Classical Models
Control Problems for Dynamical Systems Under Uncertainty and Conflict
Physics and Control in Fusion Plasma Devices
Modelling and Optimization of Beam and Plasma Dynamics
Readership: Graduate students and researchers interested in complex systems, nonlinear dynamics and experimental physics.
400pp (approx.) Pub. date: Jun 2010
ISBN: 978-981-4313-14-8
The book provides a state-of-the-art description of the construction and
properties of coherent systems on algebraic curves and their moduli spaces,
including many results based on the research works of the author and his
collaborators. This is a developing theory which generalizes the classical
theory of linear systems and has applications to higher rank Brill-Noether
theory, and projective embeddings of curves and syzygies.
Coherent Systems on Algebraic Curves begins by describing the construction of the moduli spaces and their basic properties, before proceeding on to construction methods, coherent systems in genus 0 and 1, existence of coherent systems in higher genus, irreducibility and smoothness of the moduli spaces, and special results for rank 2 and coherent systems on special curves. Many well-worked examples and open problems are included, and links with the projective geometry of curves, emphasized.
Definitions and Construction of Moduli Spaces
Basic Properties
Construction Methods and Flips
Coherent Systems in Genus 0
Coherent Systems in Genus 1
Existence of Coherent Systems in Higher Genus
Irreducibility and Smoothness of the Moduli Spaces
Special Results for Rank 2
Special Curves
Readership: Graduate students and researchers in the field of coherent systems on algebraic curves.
200pp (approx.) Pub. date: Scheduled Summer 2011
ISBN: 978-981-4304-17-7
This book begins with the basics of the geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one theorem upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify the theory and teach the reader how to apply it to solve problems in mathematics and other sciences and economics.
Each of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Brouwer's fixed point theorem, Picard's theorem and the Weierstrass approximation theorem are discussed in stared sections.
Basic Features of Euclidean Space R
Functions on Euclidean Spaces
Differential Calculus in Several Variables
Integral Calculus in Several Variables
Change of Variable Formula, Improper Multiple Integrals
Line and Surface Integrals
Elements of Ordinary and Partial Differential Equations
Introduction to Calculus of Variations
Readership: Undergraduates and academics in calculus.
450pp (approx.) Pub. date: Scheduled Winter 2010
ISBN: 978-981-4299-26-8
ISBN: 978-981-4299-27-5(pbk)
This volume consists primarily of survey papers that evolved from the lectures given in the school portion of the meeting and selected papers from the conference.
Knot theory is a very special topological subject: the classification of embeddings of a circle or collection of circles into three-dimensional space. This is a classical topological problem and a special case of the general placement problem: Understanding the embeddings of a space X in another space Y. There have been exciting new developments in the area of knot theory and 3-manifold topology in the last 25 years. From the Jones, Homflypt and Kauffman polynomials, quantum invariants of 3-manifolds, through, Vassiliev invariants, topological quantum field theories, to relations with gauge theory type invariants in 4-dimensional topology.
More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.
It is a remarkable fact that knot theory, while very special in its problematic form, involves ideas and techniques that involve and inform much of mathematics and theoretical physics. The subject has significant applications and relations with biology, physics, combinatorics, algebra and the theory of computation. The summer school on which this book is based contained excellent lectures on the many aspects of applications of knot theory. This book gives an in-depth survey of the state of the art of present day knot theory and its applications.
Readership: Researchers in knots and related topics.
Series on Knots and Everything
550pp (approx.) Pub. date: Scheduled Spring 2011
ISBN: 978-981-4307-99-4