Fedor Bogomolov and Tihomir Petrov,
New York University - Courant Institute of Mathematical Sciences, NY

Algebraic Curves and One-Dimensional Fields

Expected publication date is November 3, 2002

Description

Algebraic curves have many special properties that make their study particularly rewarding. As a result, curves provide a natural introduction to algebraic geometry. In this book, the authors also bring out aspects of curves that are unique to them and emphasize connections with algebra.

This text covers the essential topics in the geometry of algebraic curves, such as line bundles and vector bundles, the Riemann-Roch Theorem, divisors, coherent sheaves, and zeroth and first cohomology groups. The authors make a point of using concrete examples and explicit methods to ensure that the style is clear and understandable.

Several chapters develop the connections between the geometry of algebraic curves and the algebra of one-dimensional fields. This is an interesting topic that is rarely found in introductory texts on algebraic geometry.

This book makes an excellent text for a first course for graduate students.

Contents

• Algebraic Preliminaries
• From algebra to geometry
• Geometry of dimension one
• Divisors and line bundles
• Vector bundles, coherent sheaves, and cohomology
• Vector bundles on $\mathbb{P}^{1}$
• General theory of curves
• Elliptic curves
• The Riemann-Roch theorem
• Curves over arithmetic fields
• Bibliography
• Index

Details:

Series: Courant Lecture Notes, Volume: 8
Publication Year: 2002
ISBN: 0-8218-2862-2
Paging: 214 pp.
Binding: Softcover

Edited by: T. J. Lyons, University of Oxford, UK, and T. S. Salisbury, York University, Toronto, ON, Canada

Numerical Methods and Stochastics

Expected publication date is December 12, 2002

Description

This volume represents the proceedings of the Workshop on Numerical Methods and Stochastics held at The Fields Institute in April 1999. The goal of the workshop was to identify emerging ideas in probability theory that influence future work in both probability and numerical computation. The book focuses on new results and gives novel approaches to computational problems based on the latest techniques from the theory of probability and stochastic processes.

Three papers discuss particle system approximations to solutions of the stochastic filtering problem. Two papers treat particle system equations. The paper on "rough paths" describes how to generate good approximations to stochastic integrals. An expository paper discusses a long-standing conjecture: the stochastic fast dynamo effect. A final paper gives an analysis of the error in binomial and trinomial approximations to solutions of the Black-Scholes stochastic differential equations.

The book is intended for graduate students and research mathematicians interested in probability theory.

Contents

D. Crisan -- Numerical methods for solving the stochastic filtering problem
D. Crisan and T. Lyons -- Optimal filtering on discrete sets
P. Del Moral and J. Jacod -- The Monte-Carlo method for filtering with discrete-time observations: Central limit theorems
A. Guionnet -- Approximations of Markovian non linear partial differential equations by particle systems
A. Guionnet -- Non-Markovian limit diffusions and spin glasses
S. B. Hazra and F. G. Viens -- Towards pathwise stochastic fast dynamo in magneto-hydrodynamics
T. J. Lyons -- System control and rough paths
J. B. Walsh and O. D. Walsh -- Embedding and the convergence of the binomial and trinomial tree schemes

Details:

Series: Fields Institute Communications, Volume: 34
Publication Year: 2002
ISBN: 0-8218-1994-1
Paging: approximately 128 pp.
Binding: Hardcover

Ilka Agricola and Thomas Friedrich, Humboldt University, Berlin, Germany

Global Analysis: Differential Forms in Analysis, Geometry and Physics

Expected publication date is December 4, 2002

From a review of the German edition:

"Drawing on his great experience in research, writing books, teaching, and working with students, Friedrich presents once more a clearly written, smoothly readable self-contained textbook. The mathematical material and approaches are well motivated, enriched by valuable considerations and reflections. Proofs are elegant, not too technical and carefully performed ... Each chapter finishes with exercises designed to increase comprehension ... For any student who has passed the linear algebra course and calculus, this book offers an excellent opportunity to learn global analysis and its applications to mathematical physics."

-- Mathematical Reviews

Description This book is devoted to differential forms and their applications in various areas of mathematics and physics. Well-written and with plenty of examples, this introductory textbook originated from courses on geometry and analysis and presents a widely used mathematical technique in a lucid and very readable style. The authors introduce readers to the world of differential forms while covering relevant topics from analysis, differential geometry, and mathematical physics.

The book begins with a self-contained introduction to the calculus of differential forms in Euclidean space and on manifolds. Next, the focus is on Stokes' theorem, the classical integral formulas and their applications to harmonic functions and topology. The authors then discuss the integrability conditions of a Pfaffian system (Frobenius's theorem). Chapter 5 is a thorough exposition of the theory of curves and surfaces in Euclidean space in the spirit of Cartan. The following chapter covers Lie groups and homogeneous spaces. Chapter 7 addresses symplectic geometry and classical mechanics. The basic tools for the integration of the Hamiltonian equations are the moment map and completely integrable systems (Liouville-Arnold Theorem). The authors discuss Newton, Lagrange, and Hamilton formulations of mechanics. Chapter 8 contains an introduction to statistical mechanics and thermodynamics. The final chapter deals with electrodynamics. The material in the book is carefully illustrated with figures and examples, and there are over 100 exercises.

Readers should be familiar with first-year algebra and advanced calculus. The book is intended for graduate students and researchers interested in delving into geometric analysis and its applications to mathematical physics.

Contents

• Elements of multilinear algebra
  Differential forms in ${\mathbb{R}}^n$
  Vector analysis on manifolds
  Pfaffian systems
  Curves and surfaces in Euclidean 3-space
  Lie groups and homogeneous spaces
  Symplectic geometry and mechanics
  Elements of statistical mechanics and thermodynamics
  Elements of electrodynamics
  Bibliography
  Symbols
  Index

Details:

Series: Graduate Studies in Mathematics, Volume: 52
Publication Year: 2002
ISBN: 0-8218-2951-3
Paging: approximately 360 pp.
Binding: Hardcover

Henryk Iwaniec, Rutgers University, Piscataway, NJ

Spectral Methods of Automorphic Forms: Second Edition

Expected publication date is December 13, 2002

From a review of the first edition:

"The material and exposition are well-suited for second-year or higher graduate students ... This clear and comprehensive book concerning the spectral theory of $\mathrm{GL} (2)$ automorphic forms belongs on many a bookshelf."

-- Mathematical Reviews

Description

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style.

The first edition of this volume was an underground classic, both as a textbook and as a respected source for results, ideas, and references. The book's reputation sparked a growing interest in the mathematical community to bring it back into print. The AMS has answered that call with the publication of this second edition.

In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space $L^2 (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$. He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues.

Henryk Iwaniec was awarded the 2002 AMS Cole Prize for his fundamental contributions to analytic number theory. Also available from the AMS by H. Iwaniec is Topics in Classical Automorphic Forms, Volume 17 in the Graduate Studies in Mathematics series.

The book is designed for graduate students and researchers working in analytic number theory.

This book is co-published by the AMS and Revista Matematica Iberoamericana (RMI), Madrid, Spain.

Contents

• Harmonic analysis on the Euclidean plane
  Harmonic analysis on the hyperbolic plane
  Fuchsian groups
  Automorphic forms
  The spectral theorem. Discrete part
  The automorphic Green function
  Analytic continuation of the Eisenstein series
  The spectral theorem. Continuous part
  Estimates for the Fourier coefficients of Maass forms
  Spectral theory of Kloosterman sums
  The trace formula
  The distribution of eigenvalues
  Hyperbolic lattice-point problems
  Spectral bounds for cusp forms
  Classical analysis
  Special functions
  References
  Subject index
  Notation index

Details:

Series: Graduate Studies in Mathematics, Volume: 53
Publication Year: 2002
ISBN: 0-8218-3160-7
Paging: 220 pp.
Binding: Hardcover

Edited by: Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita

Integrable Systems, Topology, and Physics

Expected publication date is December 20, 2002

Description

Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the second of three collections of expository and research articles.

This volume focuses on topology and physics. The role of zero curvature equations outside of the traditional context of differential geometry has been recognized relatively recently, but it has been an extraordinarily productive one, and most of the articles in this volume make some reference to it. Symplectic geometry, Floer homology, twistor theory, quantum cohomology, and the structure of special equations of mathematical physics, such as the Toda field equations--all of these areas have gained from the integrable systems point of view and contributed to it.

Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.

The first volume from this conference also available from the AMS is Differential Geometry and Integrable Systems, Volume 308 CONM/308 in the Contemporary Mathematics series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.

Contents

• L. Casian and Y. Kodama -- Twistsed Tomei manifolds and the toda lattices
  J.-H. Chang -- Quantization of Benney hierarchies
  K. Fukaya -- Floer homology for families-A progress report
  R. Goto -- Rozansky-Witten invariants of log symplectic manifolds
  M. A. Guest -- An update on harmonic maps of finite uniton number, via the zero curvature equation
  R. Inoue -- The lattice toda field theory for simple Lie algebras
  H. Konno -- On the cohomology ring of the hyperKahler analogue of the polygon spaces
  A.-L. Mare -- On the theorem of Kim concerning $QH*(G/B)$
  Y. Nagatomo -- Geometry of the twistor equation and its applications
  A. Nakayashiki -- On the cohomology of theta divisors of hyperelliptic Jacobians
  Y. Ohyama -- Isomonodromy deformations and twistor theory
  K. Ono -- Simple singularities and symplectic fillings
  T. Otofuji -- Quantum cohomology of infinite dimensional flag manifolds
  S. Saito, N. Suzuki, and H. Yamaguchi -- Discrete conjugate nets of strings
  B. A. Shipman -- Nongeneric flows in the full Kostant-Toda lattice
  I. A. B. Strachan -- Frobenius manifolds and bi-Hamiltonian structures on discriminant hypersurfaces
  T. Taniguchi -- Periodicity conditions for harmonic maps associated to spectral data
  Y. Terashima -- Higher dimensional parallel transports for Deligne cocycles
  H.-Y. Wang -- Geometric nonlinear Schrodinger equations

Details:

Series: Contemporary Mathematics, Volume: 309
Publication Year: 2002
ISBN: 0-8218-2939-4
Paging: approximately 344 pp.
Binding: Softcover