There are a number of specialties in low-dimensional topology that can find in their family tree a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations, and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. However, its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology.

Mean curvature flow is a term that is used to describe the evolution of a hypersurface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton's theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals $pi$, the curve tends to the unit circle. In this book, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior...

This title presents articles on material from invited talks given at the IMS Workshop on Applied Probability organized by the Institute of Mathematical Sciences at the Chinese University of Hong Kong in May 1999. The goal of the workshop was to promote research in applied probability for local mathematicians and engineers and to foster exchange with experts from other parts of the world. The main themes were mathematical finance and stochastic networks.

This volume resulted from a year-long programme at the Morningside Center of Mathematics and the Academia Sinica in Beijing. It presents an overview of nonlinear conversation laws and introduces developments in this expanding field. Zhouping Xin's introductory overview of the subject is followed by lecture notes of leading experts who have made fundamental contributions to this field of research. A. Bressan's theory of $-well-posedness for entropy weak solutions to systems of nonlinear hyperbolic conversation laws in the class of viscosity solutions is one of the most important results in the...

The collection of articles in this volume are based on lectures presented during the Winter School on Mirror Symmetry held at Harvard University. There are many directions suggested by mirror symmetry which could potentially have very rich connections in physics and mathematics. This book brings together research in a major area of mathematical physics, including the progress in mirror manifolds and Lagrangian submanifolds. In particular, several articles describing homological approach and related topics are included.

This work is devoted to chaotic nonlinear dynamics. It presents a consistent, up-to-date introduction to the field of strange attractors, hyperbolic repellers, and nonlocal bifurcations. The authors keep the highest possible level of physical intuition while staying mathematically rigorous. In addition, they explain a variety of important nonstandard algorithms and problems involving the computation of chaotic dynamics. The book will help readers who are not familiar with nonlinear dynamics to understand and enjoy sophisticated modern monographs on dynamical systems and chaos. It is intended...

This volume presents lectures delivered at a workshop held at the Chinese Academy of Sciences (Bejing). The following articles are included: Nonlinear Systems Control by R. Brockett, Adaptive Control of Discrete-Time Nonlinear Systems with Structural Uncertainties by L.-L. Xie and L. Guo, Networks and Learning by P.R. Kumar, Mathematical Aspects of the Power Control Problem in Mobile Communication Systems by C.W. Sung and W.S. Wong, and Brockett's Problem on Nonlinear Filtering Theory by S.S.-T. Yau. Basic concepts and current research are both presented in the book. The volume offers a...

This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.

Discusses the differential geometric aspects of complex manifolds. This work contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. It discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles. It also gives a brief account of the surface classification theory.

The theory of Dirichlet forms brings together methods and insights from the calculus of variations, stochastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more.