N. V. Krylov, University of Minnesota, Minneapolis, MN

Introduction to the Theory of Random Processes

Expected publication date is April 11, 2002

Description
This book concentrates on some general facts and ideas of the theory of stochastic processes. The topics include the Wiener process, stationary processes, infinitely divisible processes, and Ito stochastic equations.

Basics of discrete time martingales are also presented and then used in one way or another throughout the book. Another common feature of the main body of the book is using stochastic integration with respect to random orthogonal measures. In particular, it is used for spectral representation of trajectories of stationary processes and for proving that Gaussian stationary processes with rational spectral densities are components of solutions to stochastic equations. In the case of infinitely divisible processes, stochastic integration allows for obtaining a representation of trajectories through jump measures. The Ito stochastic integral is also introduced as a particular case of stochastic integrals with respect to random orthogonal measures.

Although it is not possible to cover even a noticeable portion of the topics listed above in a short book, it is hoped that after having followed the material presented here, the reader will have acquired a good understanding of what kind of results are available and what kind of techniques are used to obtain them.

With more than 100 problems included, the book can serve as a text for an introductory course on stochastic processes or for independent study.

Other works by this author published by the AMS include, Lectures on Elliptic and Parabolic Equations in Holder Spaces and Introduction to the Theory of Diffusion Processes.

Contents

Generalities
The Wiener process
Martingales
Stationary processes
Infinitely divisible processes
Ito stochastic integral
Bibliography
Index

Details:

Series: Graduate Studies in Mathematics,Volume: 43
Publication Year: 2002
ISBN: 0-8218-2985-8
Paging: approximately 240 pp.
Binding: Hardcover

Jim Agler, University of California at San Diego, CA,
and John E. McCarthy, Washington University, St. Louis, MO

Pick Interpolation and Hilbert Function Spaces

Expected publication date is April 4, 2002

Description
The book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $H^\infty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $H^\infty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces.

The authors then consider the interpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem, and the hereditary functional calculus.

Contents

Prerequisites and notation
Introduction
Kernels and function spaces
Hardy spaces
$P^2(\mu)$
Pick redux
Qualitative properties of the solution of the Pick problem in $H^\infty(\mathbb{D})$
Characterizing kernels with the complete Pick property
The universal Pick kernel
Interpolating sequences
Model theory I: Isometries
The bidisk
The extremal three point problem on $\mathbb{D}^2$
Collections of kernels
Model theory II: Function spaces
Localization
Schur products
Parrott's lemma
Riesz interpolation
The spectral theorem for normal $m$-tuples
Bibliography
Index

Details:

Series: Graduate Studies in Mathematics,Volume: 44
Publication Year: 2002
ISBN: 0-8218-2898-3
Paging: approximately 328 pp.
Binding: Hardcover

Toshitaki Kohno, University of Tokyo, Japan

Conformal Field Theory and Topology

Expected publication date is May 11, 2002

Description
Geometry and physics have been developed with a strong influence on each other. One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. This book focuses on a relationship between two-dimensional quantum field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3-manifolds derived from Chern-Simons gauge theory. An essential difficulty in quantum field theory comes from infinite-dimensional freedom of a system. Techniques dealing with such infinite-dimensional objects developed in the framework of quantum field theory have been influential in geometry as well. This book gives an accessible treatment for a rigorous construction of topological invariants originally defined as partition functions of fields on manifolds.

The book is organized as follows: The Introduction starts from classical mechanics and explains basic background materials in quantum field theory and geometry. Chapter 1 presents conformal field theory based on the geometry of loop groups. Chapter 2 deals with the holonomy of conformal field theory. Chapter 3 treats Chern-Simons perturbation theory. The final chapter discusses topological invariants for 3-manifolds derived from Chern-Simons perturbation theory.

Contents

Geometric aspects of conformal field theory
Jones-Witten theory
Chern-Simons perturbation theory
Further developments and prospects
Bibliography
Index

Details:

Series: Translations of Mathematical Monographs,Volume: 210
Publication Year: 2002
ISBN: 0-8218-2130-X
Paging: approximately 184 pp.
Binding: Softcover

Edited by: Donald M. Davis, Lehigh University, Bethlehem, PA,
Jack Morava, Johns Hopkins University, Baltimore, MD,
Goro Nishida, Kyoto University, Japan,
W. Stephen Wilson, Johns Hopkins University, Baltimore, MD,
and Nobuaki Yagita, Ibaraki University, Japan

Recent Progress in Homotopy Theory

Expected publication date is March 14, 2002

Description
This volume presents the proceedings from the month-long program held at Johns Hopkins University (Baltimore, MD) on homotopy theory, sponsored by the Japan-U.S. Mathematics Institute (JAMI). The book centers on the following:

1. classical and nonclassical theory of $H$-spaces, compact groups, and finite groups,

2. classical and chromatic homotopy theory and localization,

3. classical and topological Hochschild cohomology,

4. elliptic cohomology and its relation to Moonshine and topological modular forms, and

5. motivic cohomology and Chow rings.

It surveys the current state of research in homotopy theory and suggests a framework for future developments. The book begins with two historical accounts, of the work of Professors Peter Landweber and Stewart Priddy, in honor of their sixtieth birthdays.

Contents
Two papers on the history of topology
H. Miller -- A marriage of manifolds and algebra: The mathematical work of Peter Landweber
N. Minami -- Some mathematical influences of Stewart Priddy
Research papers
M. Bendersky and R. D. Thompson -- Some properties of the $K$-theory completion
R. R. Bruner, D. M. Davis, and M. Mahowald -- Nonimmersions of real projective spaces implied by $tmf$
M. Brunetti -- High Euler characteristics for almost extraspecial $p$-groups
Y. Hemmi -- Unstable $p$-th order operation and $H$-spaces
M. Mahowald and M. Hopkins -- The structure of 24 dimensional manifolds having normal bundles which lift to $BO[8]$
P. Hu and I. Kriz -- The homology of $BPO$
M. Inoue -- $\mathcal{A}$-generators of the cohomology of the Steinberg summand $M(n)$
J. P. Lin -- Commutators in the homology of $H$-spaces
J. E. McClure and J. H. Smith -- A solution of Deligne's Hochschild cohomology conjecture
M. Mimura and T. Nishimoto -- Hopf algebra structure of Morava $K$-theory of the exceptional Lie groups
J. Martino and S. Priddy -- Minami-Webb type decompositions for compact Lie groups
D. C. Ravenel -- The method of infinite descent in stable homotopy theory I
K. Shimomura -- The homotopy groups $\pi_*(L_nT(m)\wedge V(n-2))$
D. Tamaki -- The fiber of iterated Freudenthal suspension and Morava K-theory of $\Omega^kS^{2\ell+1}$
M. Tanabe -- On $K$-flat and $K$-moonshine-like elements in elliptic cohomology
T. Torii -- The geometric fixed point spectrum of $({\mathbf Z}/p)^k$ Borel cohomology for $E_n$ and its completion
V. Voevodsky -- A possible new approach to the motivic spectral sequence for algebraic K-theory
C. W. Wilkerson, Jr. -- Rings of invariants and inseparable forms of algebras over the Steenrod algebra
N. Yagita -- Chow rings of classifying spaces of extraspecial $p$ groups

Details:

Series: Contemporary Mathematics,Volume: 293
Publication Year: 2002
ISBN: 0-8218-2801-0
Paging: 409 pp.
Binding: Softcover