Krantz, S.G., Washington University, St.Louis, USA,
Parks, H.R., Oregon State University, Corvallis, USA
The Implicit Function Theorem
History, Theory, and Applications
2002. Approx. 176 pages. Hardcover
ISBN 0-8176-4285-4
Due in April 2002
The implicit function theorem, part of the bedrock of mathematical analysis and geometry, has important implications in the theories of partial differential equations, differential geometry, and geometric analysis. Its history is lively and complex, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics.
There are many different forms of the implicit function theorem, including (i) the classical formulation for C++k++ functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash-Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume.
The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve.
The Implicit Function Theorem is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.
Table of contents
Preface
Introduction to the Implicit Function Theorem
History
Basic Ideas
Applications
Variations and Generalizations
Advanced Implicit Function Theorems
Glossary
Bibliography
Index
Sidoravicius, V.,
IMPA - Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil
In and Out of Equilibrium
Probability with a Physics Flavor
Progress in Probability, vol.51.
2002. Approx. 488 pages. Hardcover
ISBN 0-8176-4289-7
Due in April 2002
The intersection of probability and physics has been a rich and explosive area of growth in the past two decades. This volume contains invited articles by leading probabilists highlighting these advances. Major areas covered include: percolation theory, random walks, interacting particle systems and topics related to statistical mechanics. Graduate students and researchers in probability theory and math physics will find this book a useful reference.
In the last several years, substantial progress has been made in a number of directions: fluctuations of 2-dimensional growth processes, Wulf constructions in higher dimensions for percolation, Potts and Ising models, classification of random walks in random environments, the introduction of the stochastic Loewner equation, the rigorous proof of intersection exponents for planar Brownian motion, and finally the proof of conformal invariance for critical percolation on the triangular lattice.
This volume consists of a collection of invited articles, written by some of the most distinguished probabilists in the above-mentioned areas, most of whom were personally responsible for advances in the various subfields of probability. All of the articles are an outgrowth of the Fourth Brazilian School of Probability, held in Mambucaba, Brazil, August 2000.
Contributors: K. Alexander * J.M. Aza * J. van den Berg * T. Bodineau * F. Camia * N. Cancrini * G. Grimmett * P. Hiemer * A.E. Holroyd * H. Kesten * G.F. Lawler * T.M. Liggett * J. Lorinczi * F. Martinelli * C. M. Newman * J. Quastel * C.-E. Pfister * M. Pr"hofer * C. Roberto * O. Schramm * V. Sidoravicius * H. Spohn * A. Toom * B. T｢th * D. Ueltschi * W. Werner * M. Wschebor * M. W・hrich
Krall, A. M., The Pennsylvania State University, University Park, USA
Hilbert Space, Boundary Value Problems
and Orthogonal Polynomials
Operator Theory: Advances and Applications,vol.133.
2002. 368 pages. Hardcover
ISBN 3-7643-6701-6
English
Due in April 2002
This monograph consists of three parts: the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators; the application to linear Hamiltonian systems, giving the details of the spectral resolution; and further applications such as to orthogonal polynomials and Sobolev differential operators.
Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.
This monograph consists of three parts:
- the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators;
- the application to linear Hamiltonian systems, giving the details of the spectral resolution;
- further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.
Long, Y., Nankai University, Tianjin, People's Republic of China
Index Theory for Symplectic Paths with Applications
Progress in Mathematics, vol.207.
2002. 404 pages. Hardcover
ISBN 3-7643-6647-8
English
This is the first book that gives a systematic introduction to the index theory for symplectic matrix paths and its iteration theory, as well as applications to periodic solution problems of nonlinear Hamiltonian systems. Many topics appear for the first time in a monograph.
Researchers, graduate and postgraduate students from a wide range of areas inside mathematics or physics will benefit from this monograph that serves also as a textbook for advanced courses in symplectic geometry or Hamiltonian systems.
Among the topics covered are the algebraic and topological properties of symplectic matrices and groups, the index theory for symplectic paths, relations with other Morse-type index theories, Bott-type iteration formulae, splitting numbers, precise index iteration formulae, various index iteration inequalities, and common index properties of finitely many symplectic paths. The applications of these concepts yield new approaches to some outstanding problems and important progress on their solutions. Particular attention is given to the minimal period solution problem of Hamiltonian systems, the existence of infinitely many periodic points of the Poincare map of Lagrangian systems on tori, and the multiplicity and stability problems of closed characteristics on convex compact smooth hypersurfaces in 2n-dimensional euclidean vector space.
Valette, A., Universite de Neuchatel, Switzerland
Introduction to the Baum-Connes Conjecture
Lectures in Mathematics, ETH Zurich
2002. 116 pages. Softcover
ISBN 3-7643-6706-7
English
Due in May 2002
This book contains lecture notes on a recent conjecture that lies between algebra, geometry, topology and analysis.
It is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras.
The Baum-Connes conjecture is part of A. Connes' non-commutative geometry programme. It can be viewed as a conjectural generalisation of the Atiyah-Singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group λ).
Like the Atiyah-Singer theorem, the Baum-Connes conjecture states that a purely topological object coincides with a purely analytical one. For a given group λ, the topological object is the equivariant K-homology of the classifying space for proper actions of λ, while the analytical object is the K-theory of the C*-algebra associated with λ in its regular representation.
The Baum-Connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra. It has also strong connections with geometric group theory, as the proof of the conjecture for a given group λ usually depends heavily on geometric properties of &lamdba;.
This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. It presents, for the first time in book form, an introduction to the Baum-Connes conjecture. It starts by defining carefully the objects in both sides of the conjecture, then the assembly map which connects them. Thereafter it illustrates the main tool to attack the conjecture (Kasparov's theory), and it concludes with a rough sketch of V. Lafforgue's proof of the conjecture for co-compact lattices in in Sp(n,1), SL (3,R) and SL (3,C)