Hrsg.: Hertling, Claus / Marcolli, Matilde

Frobenius Manifolds
Quantum Cohomology and Singularities

Aspects of Mathematics E 36
Vieweg Verlag 2004. xii, 378 pp. 17 x 24 cm Hardc.
ISBN: 3-528-03206-5

Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas since the early 90's. An activity was organized at the Max-Planck-Institute for Mathematics in 2002, with the purpose of bringing together the main experts in these areas. This volume originates from this activity and presents the state of the art in the subject.

15 contributions from an activity at the MPI fur Mathematik: Complex manifolds with a multiplication and a metric on the tangent bundle. This notion was motivated by physics results. Another source of Frobenius manifolds is singularity theory. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas.

Prof. Dr. Claus Hertling, Institut fur Mathematik, Universitat Mannheim, Germany
Prof. Dr. Matilde Marcolli, Max-Planck-Institute for Mathematics, Bonn, Germany

by Kai Lai Chung (Stanford University, USA)

CHANCE AND CHOICE
Memorabilia

This book begins with a historical essay entitled “Will the Sun Rise Again?” and ends with a general address entitled “Mathematics and Applications”. The articles cover an interesting range of topics: combinatoric probabilities, classical limit theorems, Markov chains and processes, potential theory, Brownian motion, Schrodinger?Feynman problems, etc. They include many addresses presented at international conferences and special seminars, as well as memorials to and reminiscences of prominent contemporary mathematicians and reviews of their works. Rare old photos of many of them enliven the book.

Contents:

On Mutually Favorable Events
On Fluctuations in Coin-Tossing
On a Stochastic Approximation Method
On the Martin Boundary for Markov Chains
A Cluster of Great Formulas
Probabilistic Methods in Markov Chains
Markov Processes with Infinities
Probability Methods in Potential Theory
Polya’s Work in Probability
Probability and Doob
In Memory of Levy and Frechet
and other papers

Readership: Graduate students, teachers and researchers in probability and statistics.

300pp Pub. date: Scheduled Winter 2004
ISBN 981-256-012-2
ISBN 981-256-134-X(paper ed.)

by I-Hsiung Lin (National Taiwan Normal University, China)

GEOMETRIC LINEAR ALGEBRA
(Volume One)

This accessible book for beginners uses intuitive geometric concepts to create abstract algebraic theory with a special emphasis on geometric characterizations. By applying known results to describe various geometries and their invariants, the book presents problems concerned with linear algebra, such as in real and complex analysis, differential equations, differentiable manifolds, differential geometry, Markov chains and transformation groups. The clear and inductive approach makes this book unique among existing books on linear algebra both in presentation and in content.

Contents:

The Affine and Linear Structures of R1, R2 and R3:
The One-Dimensional Vector Space R (or R1)
The Two-Dimensional Vector Space R2
The Three-Dimensional Vector Space R3

Readership: Upper-level undergraduates, graduate students and lecturers.

650pp (approx.) Pub. date: Scheduled Summer 2005
ISBN 981-256-087-4
ISBN 981-256-132-3(pbk)

Alan Bundy, David Basin, Dieter Hutter, Andrew Ireland

Rippling: Meta-level Guidance for Mathematical Reasoning

220 pages 1 line diagram 10 colour figures
Hardback |
ISBN:0-521-83449-X

Rippling is a radically new technique for the automation of mathematical reasoning. It is widely applicable whenever a goal is to be proved from one or more syntactically similar givens. It was originally developed for inductive proofs, where the goal was the induction conclusion and the givens were the induction hypotheses. It has proved to be applicable to a much wider class of tasks, from summing series via analysis to general equational reasoning. The application to induction has especially important practical implications in the building of dependable IT systems, and provides solutions to issues such as the problem of combinatorial explosion. Rippling is the first of many new search control techniques based on formula annotation; some additional annotated reasoning techniques are also described here. This systematic and comprehensive introduction to rippling, and to the wider subject of automated inductive theorem proving, will be welcomed by researchers and graduate students alike.

Contents
Preface; 1. An introduction to rippling; 2. Varieties of rippling; 3. Productive use of failure; 4. A formal account of rippling; 5. The scope and limitations of rippling; 6. From rippling to a general methodology; 7. Conclusions; Appendix 1. An annotated calculus and a unification algorithm; Appendix 2. Definitions of functions used in this book; Bibliography; Index.

Peter Szekeres

A Course in Modern Mathematical Physics
Groups, Hilbert Space and Differential Geometry

Published December 2004
616 pages 48 line diagrams 341 exercises
Hardback
ISBN:0-521-82960-7 |

This book provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert space and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language. The book is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics, as well as applied mathematics. It includes numerous exercises and worked examples, to test the reader痴 understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra. No prior knowledge of group theory, abstract vector spaces or topology is required.

Contents

Preface; 1. Sets and structures; 2. Groups; 3. Vector spaces; 4. Linear operators and matrices; 5. Inner product spaces; 6. Algebras; 7. Tensors; 8. Exterior algebra; 9. Special relativity; 10. Topology; 11. Measure theory and integration; 12. Distributions; 13. Hilbert space; 14. Quantum theory; 15. Differential geometry; 16. Differentiable forms; 17. Integration on manifolds; 18. Connections and curvature; 19. Lie groups and lie algebras.

Holger Wendland

Scattered Data Approximation

346 pages 32 line diagrams 21 half-tones 53 figures
Hardback
ISBN:0-521-84335-9

Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to support courses in computer aided geometric design, and meshless methods for partial differential equations.

Contents
1. Applications and motivations; 2. Hear spaces and multivariate polynomials; 3. Local polynomial reproduction; 4. Moving least squares; 5. Auxiliary tools from analysis and measure theory; 6. Positive definite functions; 7. Completely monotine functions; 8. Conditionally positive definite functions; 9. Compactly supported functions; 10. Native spaces; 11. Error estimates for radial basis function interpolation; 12. Stability; 13. Optimal recovery; 14. Data structures; 15. Numerical methods; 16. Generalized interpolation; 17. Interpolation on spheres and other manifolds.

I. Assem, A. Skowronski, D. Simson

Elements of the Representation Theory of Associative Algebras, Volume 1
Techniques of Representation Theory

472 pages
Paperback |Dec.2004 ISBN:0-521-58631-3
Hardback |Dec.2004 ISBN:0-521-58423-X

This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

Contents

0. Introduction; 1. Algebras and modules; 2. Quivers and algebras; 3. Representations and modules; 4. Auslander-Reiten theory; 5. Nakayama algebras and representation-finite group algebras; 6. Tilting theory; 7. Representation-finite hereditary algebras; 8. Tilted algebras; 9. Directing modules and postprojective components; A. Appendix. Categories functors and homology.