Series: Universitext
2008, Approx. 295 p., Softcover
ISBN: 978-1-84800-053-7
Due: January 2008
About this textbook
This book provides an introduction to algebraic geometric methods in the theory of complex linear differential equations. Starting from basic notions in complex algebraic geometry, it develops some of the classical problems of linear differential equations and ends with applications to recent research questions related to mirror symmetry.
The fundamental tool used in this book is that of a vector bundle with connection. A detailed analysis of the singularities of such objects (regular or irregular singularities, the Riemann-Hilbert problem), and of their deformations, is worked out. The notion of a Frobenius structure on a complex analytic manifold appeared at the end of the seventies in the theory of singularities of holomorphic functions. Motivated by physical considerations, it has concerned other mathematical domains. The book also develops an approach to Frobenius manifolds using isomonodromic deformations of linear differential equations.
Series: Progress in Mathematics , Vol. 265
2008, Approx. 800 p., Hardcover
ISBN: 978-3-7643-8607-8
Due: December 2007
About this book
This book presents 18 articles by prominent mathematicians, dedicated to the memory of Alexander Reznikov (1960-2003), a brilliant highly original mathematician with broad mathematical interests. In addition it contains an influential, so far unpublished manuscript of Reznikov of book length. The research articles broadly reflect the range of Reznikov's own interests in geometry, group and number theory, functional analysis, dynamical systems and topology. In addition, there are surveys "Geometrization of probability", "Kleinian groups in higher dimensions", "(C,F)-construction of funny rank-one actions for locally compact groups", and some articles centering on Reznikov as a person.
Table of contents
Preface.- Alexander Reznikov (1960-2003).- Contributions by V. Armitage, J.-M. Bismut, D.V. Borisov, P. Bressler, A. Danilenko, T. Delzant, H. Esnault, A. Fel'shtyn, D. Goncalves, A. Goncharov, A. Gorokhovsky, R. Grigorchuk, B. Guilfoyle, P.H. Hai, J.N. Iyer, M. Kapovich, W. Klingenberg, V. Lyubashenko, Y.I. Manin, O. Manzyuk, V.T. Milman, P. Moree, M. Morishita, R. Nest, J.-P. Otal, J.R. Parker, A. Reznikov, C.T. Simpson, Y. Soibelman, X. Sun, B. Tsygan, M. Zaidenberg
Series: Universitext
2008, Approx. 320 p., Softcover
ISBN: 978-0-387-74532-9
Due: January 2008
About this textbook
Introduces hyperbolic geometry from an axiomatic viewpoint
Includes a wide-ranging selection of unique topics, such as the Lorentz group and special relativity
Well-written and self-contained, ideally suited for use in the classroom
Heavily illustrated
This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity.
Changes to this second edition include: new chapters on Euclidean geometry, as a means of comparing Euclidean and hyperbolic geometry; a new chapter on Euclidean planes; a new proof of the uniqueness of the hyperbolic plane (up to isomorphism and choice of unit length); a proof that the upper half-plane can be made a hyperbolic plane with a new way to measure distance; a new appendix on analysis; the addition of new theorems and the improvement of existing ones.
Table of contents
Preface.- Introduction.- Axioms for Plane Geometry.- Euclidean Geometry.- Euclidean Planes.- Some Neutral Theorems of Plane Geometry.- Qualitative Description of the Hyperbolic Plane.- H3 and Euclidean Approximations in H2.- Differential Geometry of Surface.- Quantitative Considerations.- Consistency and Categoricalness of the Hyperbolic Axioms- the Classical Models.- Matrix Representation of the Isometry Group.- Differential and Hyperbolic Geometry in More Dimensions.- Connections with the Lorentz Group of Special Relativity.- Constructions by Straightedge and Compass in the Hyperbolic Plane.- Index
Series: Frontiers in Mathematics
2008, Approx. 200 p., Softcover
ISBN: 978-3-7643-8613-9
Due: March 2008
About this book
This book arose out of original researches of the authors on the extension of well-established applications of complex numbers, related to Euclidean geometry, to the space-time symmetry of two-dimensional Special Relativity.
The system of hyperbolic numbers (the simplest extension of complex numbers) is extensively studied and applied to to typical instances as the ‘’twin-paradox’’ and a plain exposition of space-time geometry and trigonometry is given.
The application of hyperbolic numbers to Special Relativity suggest trying the possible application of multidimensional hypercomplex systems.
Commutative hypercomplex systems with four unities are studied and attention is drown on their interesting properties.
Written for:
Mathematicians, physicists, engineers and interdisciplinary researchers in mathematical physics or related fields; to some extent also suitable for lectures on a graduate level
Keywords:
Lorentz surfaces
Minkowski space-time
gauss differential geometry
hyperbolic numbers
hypercomplex numbers
special relativity
twin paradox
Series: Monografie Matematyczne , Vol. 69
2008, Approx. 350 p., Hardcover
ISBN: 978-3-7643-8601-6
Due: March 2008
About this book
This volume is a thorough and comprehensive treatise on vector measures. The functions to be integrated can be either [0,infinity]- or real- or complex-valued and the vector measure can take its values in arbitrary locally convex Hausdorff spaces. Moreover, the domian of the vector measure does not have to be a sigma-algebra: it can also be a delta-ring. The book contains not only a large amount of new material but also corrects various errors in well-known results available in the literature. It will appeal to a wide audience of mathematical analysts.
Table of contents
Preface.- 1. Preliminaries.- 2. Basic Properties of the Bartle-Dunford-Schwartz Integral.- 3. Lp-Spaces.- 4. Integration with Respect to LcHs-valued Measures.- 5. Applications to Integration in Locally Compact Hausdorff Spaces I.- 6. Applications to Integration in Locally Compact Hausdorff Spaces II.- 7. Complements to the Thomas Theory of Vectorial Radon Integration.- References.- List of symbols.- Index.