Richard Montgomery, University of California, Santa Cruz, CA

A Tour of Subriemannian Geometries,
Their Geodesics and Applications

Description
Subriemannian geometries, also known as Carnot-Caratheodory geometries, can be viewed as limits of Riemannian geometries. They also arise in physical phenomenon involving "geometric phases" or holonomy. Very roughly speaking, a subriemannian geometry consists of a manifold endowed with a distribution (meaning a $k$-plane field, or subbundle of the tangent bundle), called horizontal together with an inner product on that distribution. If $k=n$, the dimension of the manifold, we get the usual Riemannian geometry. Given a subriemannian geometry, we can define the distance between two points just as in the Riemannin case, except we are only allowed to travel along the horizontal lines between two points.

The book is devoted to the study of subriemannian geometries, their geodesics, and their applications. It starts with the simplest nontrivial example of a subriemannian geometry: the two-dimensional isoperimetric problem reformulated as a problem of finding subriemannian geodesics. Among topics discussed in other chapters of the first part of the book we mention an elementary exposition of Gromov's surprising idea to use subriemannian geometry for proving a theorem in discrete group theory and Cartan's method of equivalence applied to the problem of understanding invariants (diffeomorphism types) of distributions. There is also a chapter devoted to open problems.

The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail the following four physical problems: Berry's phase in quantum mechanics, the problem of a falling cat righting herself, that of a microorganism swimming, and a phase problem arising in the $N$-body problem. He shows that all these problems can be studied using the same underlying type of subriemannian geometry: that of a principal bundle endowed with $G$-invariant metrics.

Reading the book requires introductory knowledge of differential geometry, and it can serve as a good introduction to this new exciting area of mathematics.

Contents
Geodesics in subriemannian manifolds
Dido meets Heisenberg
Chow's theorem: Getting from A to B
A remarkable horizontal curve
Curvature and nilpotentization
Singular curves and geodesics
A zoo of distributions
Cartan's approach
The tangent cone and Carnot groups
Discrete groups tending to Carnot geometries
Open problems
Mechanics and geometry of bundles
Metrics on bundles
Classical particles in Yang-Mills fields
Quantum phases
Falling, swimming, and orbiting
Appendices
Geometric mechanics
Bundles and the Hopf fibration
The Sussmann and Ambrose-Singer theorems
Calculus of the endpoint map and existence of geodesics
Bibliography
Index

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical Surveys and Monographs,Volume: 91
Publication Year: 2002
ISBN: 0-8218-1391-9
Paging: 259 pp.
Binding: Hardcover

Edited by: Joshua A. Leslie and Thierry P. Robart,
Howard University, Washington, DC

The Geometrical Study of Differential Equations

Description
This volume contains papers based on some of the talks given at the NSF-CBMS conference on "The Geometrical Study of Differential Equations" held at Howard University (Washington, DC). The collected papers present important recent developments in this area, including the treatment of nontransversal group actions in the theory of group invariant solutions of PDEs, a method for obtaining discrete symmetries of differential equations, the establishment of a group-invariant version of the variational complex based on a general moving frame construction, the introduction of a new variational complex for the calculus of difference equations and an original structural investigation of Lie-Backlund transformations. The book opens with a modern and illuminating overview of Lie's line-sphere correspondence and concludes with several interesting open problems arising from symmetry analysis of PDEs. It offers a rich source of inspiration for new or established researchers in the field.

This book can serve nicely as a companion volume to a forthcoming book written by the principle speaker at the conference, Professor Niky Kamran, to be published in the AMS series, CBMS Regional Conference Series in Mathematics.

Contents

R. Milson -- An overview of Lie's line-sphere correspondence
V. Torrisi and M. C. Nucci -- Application of Lie group analysis to a mathematical model which describes HIV transmission
R. Beals -- Geometry and PDE on the Heisenberg group: A case study
G. Mari Beffa -- Invariant evolutions of curves and surfaces and completely integrable Hamiltonian systems
B. A. Shipman -- On the fixed points of the toda hierarchy
I. M. Anderson, M. E. Fels, and C. G. Torre -- Group invariant solutions in mathematical physics and differential geometry
P. E. Hydon -- Discrete symmetries of differential equations
T. A. Ivey -- Integrable geometric evolution equations for curves
J. A. Sanders and J. P. Wang -- On integrability of evolution equations and representation theory
M. Oberguggenberger -- Symmetry groups, nonlinear partial differential equations, and generalized functions
R. H. Heredero -- Lie symmetries of differential-difference equations
E. L. Mansfield and P. E. Hydon -- On a variational complex for difference equations
I. A. Kogan and P. J. Olver -- The invariant variational bicomplex
E. G. Reyes -- On geometrically integrable equations and hierarchies of pseudo-spherical type
I. A. Kogan -- Inductive construction of moving frames
V. Itskov -- Orbit reduction of contact ideals and group-invariant variational problems
T. Robart -- About the local and formal geometry of PDE
P. A. Clarkson and E. L. Mansfield -- Open problems in symmetry analysis


Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics,Volume: 285
Publication Year: 2001
ISBN: 0-8218-2964-5
Paging: 205 pp.
Binding: Softcover

Edited by: Marlos A. G. Viana, University of Illinois at Chicago, IL,
and Donald St. P. Richards, University of Virginia, Charlottesville, VA

Algebraic Methods in Statistics and Probability

Description
Algebraic methods and arguments in statistics and probability are well known, from Gauss's least squares principle through Fisher's method of variance decomposition. The relevance of group-theoretic arguments, for example, became evident in the 1980s. Such techniques continue to be of interest today, along with other developments, such as the use of graph theory in modelling complex stochastic systems.

This volume is based on lectures presented at the AMS Special Session on Algebraic Methods and Statistics held at the University of Notre Dame (Indiana) and on contributed articles solicited for this volume. The articles are intended to foster communication between representatives of the diverse scientific areas in which these functions are utilized and to further the trend of utilizing algebraic methods in the areas of statistics and probability.

This is one of few volumes devoted to the subject of algebraic methods in statistics and probability. The wide range of topics covered in this volume demonstrates the vigorous level of research and opportunities ongoing in these areas.

Contents

J. Aitchison -- Simplicial inference
J.-F. Burnol -- A note on Nyman's equivalent formulation of the Riemann hypothesis
D. Collombier and A. Jourdan -- On the construction of linear orthogonal arrays by extension
A. Di Bucchianico and D. E. Loeb -- A coordinate-free approach to multivariate exponential families
M. L. Eaton and W. D. Sudderth -- Best invariant predictive distributions
W. Ehm -- A family of probability densities related to the Riemann zeta function
S. N. Evans -- Local field $U$-statistics
P. Feinsilver and J. Kocik -- Krawtchouk matrices from classical and quantum random walks
Y. Gao and J. I. Marden -- Some rank-based hypothesis tests for covariance structure and conditional independence
P. Graczyk -- Gaussian measures as limits on irreducible symmetric spaces and cones
R. D. Gupta and D. St. Richards -- The covariance structure of the multivariate Liouville distributions
I. S. Helland -- Reduction of regression models under symmetry
P. T. Kim and D. St. Richards -- Deconvolution density estimation on compact Lie groups
C. A. J. Klaassen, E.-J. Lee, and F. H. Ruymgaart -- On efficiency of indirect estimation of nonparametric regression functions
T. Kollo and D. von Rosen -- Patterned matrices treated via linear spaces
S. P. Lalley -- Random walks on regular languages and algebraic systems of generating functions
G. Letac and H. Massam -- The normal quasi-Wishart distribution
T. Neeman and T. Chang -- Rank score statistics for spherical data
M. D. Perlman -- Graphical model search via essential graphs
G. Pistone, E. Riccomagno, and H. P. Wynn -- Computational commutative algebra in discrete statistics
A. Takemura and S. Kuriki -- Maximum covariance difference test for equality of two covariance matrices
M. A. G. Viana -- The covariance structure of random permutation matrices
E. Wit and P. McCullagh -- The extendibility of statistical models

Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics,Volume: 287
Publication Year: 2002
ISBN: 0-8218-2687-5
Paging: 340 pp.
Binding: Softcover

Edited by: Salvador Perez-Esteva and Carlos Villegas-Blas,
Both Universidad Nacional Autonoma de Mexico

Second Summer School in Analysis and Mathematical Physics: Topics in Analysis: Harmonic, Complex, Nonlinear and Quantization

Description
For the second time, a Summer School in Analysis and Mathematical Physics took place at the Universidad Nacional Autonoma de Mexico in Cuernavaca. The purpose of the schools is to provide a bridge from standard graduate courses in mathematics to current research topics, particularly in analysis. The lectures are given by internationally recognized specialists in the fields. The topics covered in this Second Summer School include harmonic analysis, complex analysis, pseudodifferential operators, the mathematics of quantum chaos, and non-linear analysis.

This volume is a joint publication of the American Mathematical Society and the Sociedad Matematica Mexicana.

Contents

M. C. Pereyra -- Lecture notes on dyadic harmonic analysis
S. Hofmann -- A short course on the Kato problem
N. L. Vasilevski -- Toeplitz operators on the Bergman spaces: Inside-the-domain effects
Y. V. Egorov -- Pseudo-differential operators and quantization
S. De Bievre -- Quantum chaos: A brief first visit
P. Padilla -- Variational methods in nonlinear analysis
J. Jacobsen -- A globalization of the implicit function theorem with applications to nonlinear elliptic equations

Details:

Publisher: American Mathematical Society, Sociedad Matematica Mexicana
Distributor: American Mathematical Society
Series: Contemporary Mathematics,Volume: 289
Publication Year: 2001
ISBN: 0-8218-2708-1
Paging: 272 pp.
Binding: Softcover

Jin Hong and Seok-Jin Kang, Korea Institute for Advanced Study, Seoul, Korea

Introduction to Quantum Groups and Crystal Bases

Description
The notion of a "quantum group" was introduced by V.G. Dinfeld and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras.

In particular, the theory of "crystal bases" or "canonical bases" developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.

The authors start with the basic theory of quantum groups and their representations, and then give a detailed exposition of the fundamental features of crystal basis theory. They also discuss its applications to the representation theory of classical Lie algebras and quantum affine algebras, solvable lattice model theory, and combinatorics of Young walls.

Contents

Lie algebras and Hopf algebras
Kac-Moody algebras
Quantum groups
Crystal bases
Existence and uniqueness of crystal bases
Global bases
Young tableaux and crystals
Crystal graphs for classical Lie algebras
Solvable lattice models
Perfect crystals
Combinatorics of young walls
Bibliography
Index of symbols
Index


Details:

Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Graduate Studies in Mathematics,
Publication Year: 2002
ISBN: 0-8218-2874-6
Paging: approximately 328 pp.
Binding: Hardcover