The action of a compact Lie group, G, on a compact sympletic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytopes, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope. This book is addressed to researchers and can be used as a semester text.

The geometry of lines occurs naturally in such different areas as sculptured surface machining, computation of offsets and medial axes, surface reconstruction for reverse engineering, geometrical optics, kinematics and motion design, and modeling of developable surfaces. This book covers line geometry from various viewpoints and aims towards computation and visualization. Besides applications, it contains a tutorial on projective geometry and an introduction into the theory of smooth and algebraic manifolds of lines. It will be useful to researchers, graduate students, and anyone interested either in the theory or in computational aspects in general, or in applications in particular.

A major flaw in semi-Riemannian geometry is a shortage of suitable types of maps between semi-Riemannian manifolds that will compare their geometric properties. Here, a class of such maps called semi-Riemannian maps is introduced. The main purpose of this book is to present results in semi-Riemannian geometry obtained by the existence of such a map between semi-Riemannian manifolds, as well as to encourage the reader to explore these maps. The first three chapters are devoted to the development of fundamental concepts and formulas in semi-Riemannian geometry which are used throughout the work. In Chapters 4 and 5 semi-Riemannian maps and such maps with respect to a semi-Riemannian foliation are studied. Chapter 6 studies the maps from a semi-Riemannian manifold to 1-dimensional semi- Euclidean space. In Chapter 7 some splitting theorems are obtained by using the existence of a semi-Riemannian map. Audience: This volume will be of interest to mathematicians and physicists whose work involves differential geometry, global analysis, or relativity and gravitation.

A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam ple, partially ordered groups with interpolation property were intro duced in F. Riesz's fundamental paper 1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P."

In this book the authors describe the important generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La conjecture de Weil II". The authors follow the important and beautiful methods of Laumon and Brylinski which lead to a simplification of Deligne's theory. Deligne's work is closely related to the sheaf theoretic theory of perverse sheaves. In this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore the authors include the complete theory of middle perverse sheaves. In this part, the l-adic Fourier transform is introduced as a technique providing natural and simple proofs. To round things off, there are three chapters with significant applications of these theories.

This volume contains research and survey articles by well known and respected mathematicians on differential geometry and topology that have been collected and dedicated in honor of Lieven Vanhecke, as a tribute to his many fruitful and inspiring contributions to these fields.

The papers, all written with the necessary introductory and contextual material, describe recent developments and research trends in spectral geometry, the theory of geodesics and curvature, contact and symplectic geometry, complex geometry, algebraic topology, homogeneous and symmetric spaces, and various applications of partial differential equations and differential systems to geometry. One of the key strengths of these articles is their appeal to non-specialists, as well as researchers and differential geometers.

Contributors: D.E. Blair; E. Boeckx; A.A. Borisenko; G. Calvaruso; V. CortA(c)s; P. de Bartolomeis; J.C. DA-az-Ramos; M. Djoric; C. Dunn; M. FernAndez; A. Fujiki; E. GarcA-a-RA-o; P.B. Gilkey; O. Gil-Medrano; L. Hervella; O. Kowalski; V. MuAoz; M. Pontecorvo; A.M. Naveira; T. Oguro; L. SchAfer; K. Sekigawa; C-L. Terng; K. Tsukada; Z. VlAAek; E. Wang; and J.A. Wolf.

As many readers will know, the 20th century was a time when the fields of mathematics and the sciences were seen as two separate entities. Caused by the rapid growth of the physical sciences and an increasing abstraction in mathematical research, each party, physicists and mathematicians alike, suffered a misconception; not only of the opposition's theoretical underpinning, but of how the two subjects could be intertwined and effectively utilized. One sub-discipline that played a part in the union of the two subjects is Theoretical Physics. Breaking it down further came the fundamental theories, Relativity and Quantum theory, and later on Yang-Mills theory. Other areas to emerge in this area are those derived from the works of Donaldson, Chern-Simons, Floer-Fukaya, and Seiberg-Witten. Aimed at a wide audience, Physical Topics in Mathematics demonstrates how various physical theories have played a crucial role in the developments of Mathematics and in particular, Geometric Topology. Issues are studied in great detail, and the book steadfastly covers the background of both Mathematics and Theoretical Physics in an effort to bring the reader to a deeper understanding of their interaction. Whilst the world of Theoretical Physics and Mathematics is boundless; it is not the intention of this book to cover its enormity. Instead, it seeks to lead the reader through the world of Physical Mathematics; leaving them with a choice of which realm they wish to visit next.

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, geometrical constructions and finite field extensions, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra. The text is intended for junior- to senior-level mathematics majors. Robin Hartshorne is a professor of mathematics at the University of California at Berkeley, and is the author of Foundations of Projective Geometry (Benjamin, 1967) and Algebraic Geometry (Springer, 1977).

The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres ponding periodic orbit in the geodesic flow is infinitesimally stable.

curvilinear coordinates. This treatment includes in particular a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The fourth and last chapter, which heavily relies on Chapter 2, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are "two-dimensional," in the sense that they are expressed in terms of two curvilinear coordinates used for de?ning the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental "Korn inequality on a surface" and to an "in?nit- imal rigid displacement lemma on a surface." This chapter also includes a brief introduction to other two-dimensional shell equations. Interestingly, notions that pertain to di?erential geometry per se, suchas covariant derivatives of tensor ?elds, are also introduced in Chapters 3 and 4, where they appear most naturally in the derivation of the basic boundary value problems of three-dimensional elasticity and shell theory. Occasionally, portions of the material covered here are adapted from - cerpts from my book "Mathematical Elasticity, Volume III: Theory of Shells," published in 2000by North-Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to rely on such excerpts. Oth- wise, the bulk of this work was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].

Riemannian Topology and Structures on Manifolds results from a similarly entitled conference held on the occasion of Charles P. Boyer s 65th birthday. The various contributions to this volume discuss recent advances in the areas of positive sectional curvature, Kahler and Sasakian geometry, and their interrelation to mathematical physics, especially M and superstring theory. Focusing on these fundamental ideas, this collection presents review articles, original results, and open problems of interest. "

"Geometry and Physics" addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jurgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective."

This text describes how fractal phenomena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phenomena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phenomena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cannot serve for modeling complex phenomena; "Physics of Fractal Operators" shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long known not to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Physics of Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nonlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phenomena become more and more complicated.

A novel feature of the book is its integrated approach to algebraic surface theory and the study of vector bundle theory on both curves and surfaces. While the two subjects remain separate through the first few chapters, they become much more tightly interconnected as the book progresses. Thus vector bundles over curves are studied to understand ruled surfaces, and then reappear in the proof of Bogomolov's inequality for stable bundles, which is itself applied to study canonical embeddings of surfaces via Reider's method. Similarly, ruled and elliptic surfaces are discussed in detail, before the geometry of vector bundles over such surfaces is analysed. Many of the results on vector bundles appear for the first time in book form, backed by many examples, both of surfaces and vector bundles, and over 100 exercises forming an integral part of the text. Aimed at graduates with a thorough first-year course in algebraic geometry, as well as more advanced students and researchers in the areas of algebraic geometry, gauge theory, or 4-manifold topology, many of the results on vector bundles will also be of interest to physicists studying string theory.

This is the first book in the Selecta, the collected works of Benoit Mandelbrot. This volume incorporates his original contributions to finance. The chapters consist of much new material prepared for this volume, as well as reprints of his classic papers which are devoted to the roles that discontinuity and related forms of concentration play in finance and economics. Much of this work helps to lay a foundation for evaluating risks in trading strategies.

Shape interrogation is the process of extraction of information from a geometric model. It is a fundamental component of Computer Aided Design and Manufacturing (CAD/CAM) systems. The authors focus on shape interrogation of geometric models bounded by free-form surfaces. Free-form surfaces, also called sculptured surfaces, are widely used in the bodies of ships, automobiles and aircraft, which have both functionality and attractive shape requirements. Many electronic devices as well as consumer products are designed with aesthetic shapes, which involve free-form surfaces. This book provides the mathematical fundamentals as well as algorithms for various shape interrogation methods including nonlinear polynomial solvers, intersection problems, differential geometry of intersection curves, distance functions, curve and surface interrogation, umbilics and lines of curvature, geodesics, and offset curves and surfaces. This book will be of interest both to graduate students and professionals.

This book presents methods of solving problems in three areas of elementary combinatorial mathematics: classical combinatorics, combinatorial arithmetic, and combinatorial geometry. In each topic, brief theoretical discussions are immediately followed by carefully worked-out examples of increasing degrees of difficulty, and by exercises that range from routine to rather challenging. While this book emphasizes some methods that are not usually covered in beginning university courses, it nevertheless teaches techniques and skills that are useful not only in the specific topics covered here. There are approximately 310 examples and 650 exercises. Jiri Herman is the headmaster of a prestigious secondary school (Gymnazium) in Brno, Radan Kucera is Associate Professor of Mathematics at Masaryk University in Brno, and Jaromir Simsa is a researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic. The translator, Karl Dilcher, is Professor of Mathematics at Dalhousie University in Canada. This book can be seen as a continuation of the previous book by the same authors and also translated by Karl Dilcher, Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory (Springer-Verlag 2000).

With a focus on the interplay between mathematics and applications of imaging, the first part covers topics from optimization, inverse problems and shape spaces to computer vision and computational anatomy. The second part is geared towards geometric control and related topics, including Riemannian geometry, celestial mechanics and quantum control. Contents: Part I Second-order decomposition model for image processing: numerical experimentation Optimizing spatial and tonal data for PDE-based inpainting Image registration using phase amplitude separation Rotation invariance in exemplar-based image inpainting Convective regularization for optical flow A variational method for quantitative photoacoustic tomography with piecewise constant coefficients On optical flow models for variational motion estimation Bilevel approaches for learning of variational imaging models Part II Non-degenerate forms of the generalized Euler Lagrange condition for state-constrained optimal control problems The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls Controllability of Keplerian motion with low-thrust control systems Higher variational equation techniques for the integrability of homogeneous potentials Introduction to KAM theory with a view to celestial mechanics Invariants of contact sub-pseudo-Riemannian structures and Einstein Weyl geometry Time-optimal control for a perturbed Brockett integrator Twist maps and Arnold diffusion for diffeomorphisms A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I Index

Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications, it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics.

The transparency and power of geometric constructions has been a source of inspiration to generations of mathematicians. The beauty and persuasion of pictures, communicated in words or drawings, continues to provide the intuition and arguments for working with complicated concepts and structures of modern mathematics. This volume contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory.

Key topics include:

* Curves and their Jacobians

* Algebraic surfaces

* Moduli spaces, Shimura varieties

* Motives and motivic integration

* Number-theoretic applications, rational points

* Combinatorial aspects of algebraic geometry

* Quantum cohomology

* Arithmetic dynamical systems

The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry; the text can serve as an intense introduction for graduate students and those wishing to pursue research in these areas.

Contributors: I. Bauer, F. Bogomolov, N. Budur, F. Catanese, C.-L. Chai, R. Cluckers, C. De Concini, J.S. Ellenberg, F. Grunewald, B. Hassett, T. Hausel, F. Loeser, J. Pineiro, R. Pink, C. Procesi, M. Spitzweck, P. Swinnerton-Dyer, L. Szpiro, H. Tamvakis, Y. Tschinkel, T.J. Tucker, A. Venkatesh, and Y.G. Zarhin.

Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics. The three volumes of Modern Geometry - Methods and Applications contain a concrete exposition of these methods together with their main applications in mathematics and physics. This third volume, presented in highly accessible languages, concentrates in homology theory. It contains introductions to the contemporary methods for the calculation of homology groups and the classification of manifesto. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

Ordinary differential control thPory (the classical theory) studies input/output re lations defined by systems of ordinary differential equations (ODE). The various con cepts that can be introduced (controllability, observability, invertibility, etc. ) must be tested on formal objects (matrices, vector fields, etc. ) by means of formal operations (multiplication, bracket, rank, etc. ), but without appealing to the explicit integration (search for trajectories, etc. ) of the given ODE. Many partial results have been re cently unified by means of new formal methods coming from differential geometry and differential algebra. However, certain problems (invariance, equivalence, linearization, etc. ) naturally lead to systems of partial differential equations (PDE). More generally, partial differential control theory studies input/output relations defined by systems of PDE (mechanics, thermodynamics, hydrodynamics, plasma physics, robotics, etc. ). One of the aims of this book is to extend the preceding con cepts to this new situation, where, of course, functional analysis and/or a dynamical system approach cannot be used. A link will be exhibited between this domain of applied mathematics and the famous 'Backlund problem', existing in the study of solitary waves or solitons. In particular, we shall show how the methods of differ ential elimination presented here will allow us to determine compatibility conditions on input and/or output as a better understanding of the foundations of control the ory. At the same time we shall unify differential geometry and differential algebra in a new framework, called differential algebraic geometry."

This well-organized and coherent collection of papers leads the reader to the frontiers of  present research in the theory of nonlinear partial differential equations and the calculus of variations and offers insight into some exciting developments. In addition, most articles also provide an excellent introduction to their background, describing extensively as they do the history of those problems presented, as well as the state of the art and offer a well-chosen guide to the literature. Part I contains the contributions of geometric nature: From spectral theory on regular and singular spaces to regularity theory of solutions of variational problems. Part II consists of articles on partial differential equations which originate from problems in physics, biology and stochastics. They cover elliptic, hyperbolic and parabolic cases.

Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.