Providing an accessible approach to a special case of the Rank Theorem, the present text considers the exact finiteness properties of S-arithmetic subgroups of split reductive groups in positive characteristic when S contains only two places. While the proof of the general Rank Theorem uses an involved reduction theory due to Harder, by imposing the restrictions that the group is split and that S has only two places, one can instead make use of the theory of twin buildings.

The characteristic feature of many models for field theories based on concepts of differential geometry is their nonlinearity. In this book a systematic exposition of nonlinear transformations in quantum field theory is given. The book starts with a short account of the renormalization theory with examples which can be handled successfully in four space-time dimensions. The second part is devoted to nonlinear sigma-models and their constructions in two dimensions. In the final section geometrical and cohomological methods and the relations to string theory are treated. This book is an important contribution towards rigorous definitions, and the mastering of nonlinear reparametrizations in agreement with the principles of quantum field theory will help to deal with anomalies, geometry and the like consistently and thus to understand better their implications for physics. The collection of papers addresses researchers and graduate students as well and will stimulate further work on the foundations of quantum field theory.

This two-part volume contains numerous examples and insights on various topics. The authors have taken pains to present the material rigorously and coherently. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

This book introduces the reader to powerful methods of critical point theory and details successful contemporary approaches to many problems, some of which had proved resistant to attack by older methods. Topics covered include Morse theory, critical groups, the minimax principle, various notions of linking, jumping nonlinearities and the Fucik spectrum in an abstract setting, sandwich pairs and the cohomological index. Applications to semilinear elliptic boundary value problems, p-Laplacian problems and anisotropic systems are given. Written for graduate students and research scientists, the book includes numerous examples and presents more recent developments in the subject to bring the reader up to date with the latest research.

This is the Proceedings of the ICM 2010 Satellite Conference on "Buildings, Finite Geometries and Groups" organized at the Indian Statistical Institute, Bangalore, during August 29 - 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups.

The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

The present work grew out of a study of the Maslov class (e. g. (37]), which is a fundamental invariant in asymptotic analysis of partial differential equations of quantum physics. One of the many in terpretations of this class was given by F. Kamber and Ph. Tondeur (43], and it indicates that the Maslov class is a secondary characteristic class of a complex trivial vector bundle endowed with a real reduction of its structure group. (In the basic paper of V. I. Arnold about the Maslov class (2], it is also pointed out without details that the Maslov class is characteristic in the category of vector bundles mentioned pre viously. ) Accordingly, we wanted to study the whole range of secondary characteristic classes involved in this interpretation, and we gave a short description of the results in (83]. It turned out that a complete exposition of this theory was rather lengthy, and, moreover, I felt that many potential readers would have to use a lot of scattered references in order to find the necessary information from either symplectic geometry or the theory of the secondary characteristic classes. On the otherhand, both these subjects are of a much larger interest in differential geome try and topology, and in the applications to physical theories."

In the past ten years, there has been much progress in understanding the global dynamics of systems with several degrees-of-freedom. An important tool in these studies has been the theory of normally hyperbolic invariant manifolds and foliations of normally hyperbolic invariant manifolds. In recent years these techniques have been used for the development of global perturbation methods, the study of resonance phenomena in coupled oscillators, geometric singular perturbation theory, and the study of bursting phenomena in biological oscillators. "Invariant manifold theorems" have become standard tools for applied mathematicians, physicists, engineers, and virtually anyone working on nonlinear problems from a geometric viewpoint. In this book, the author gives a self-contained development of these ideas as well as proofs of the main theorems along the lines of the seminal works of Fenichel. In general, the Fenichel theory is very valuable for many applications, but it is not easy for people to get into from existing literature. This book provides an excellent avenue to that. Wiggins also describes a variety of settings where these techniques can be used in applications.

The flood of information through various computer networks such as the In ternet characterizes the world situation in which we live. Information worlds, often called virtual spaces and cyberspaces, have been formed on computer networks. The complexity of information worlds has been increasing almost exponentially through the exponential growth of computer networks. Such nonlinearity in growth and in scope characterizes information worlds. In other words, the characterization of nonlinearity is the key to understanding, utiliz ing and living with the flood of information. The characterization approach is by characteristic points such as peaks, pits, and passes, according to the Morse theory. Another approach is by singularity signs such as folds and cusps. Atoms and molecules are the other fundamental characterization ap proach. Topology and geometry, including differential topology, serve as the framework for the characterization. Topological Modeling for Visualization is a textbook for those interested in this characterization, to understand what it is and how to do it. Understanding is the key to utilizing information worlds and to living with the changes in the real world. Writing this textbook required careful preparation by the authors. There are complex mathematical concepts that require designing a writing style that facilitates understanding and appeals to the reader. To evolve a style, we set as a main goal of this book the establishment of a link between the theoretical aspects of modern geometry and topology, on the one hand, and experimental computer geometry, on the other.

Providing an up-to-date overview of the geometry of manifolds with non-negative sectional curvature, this volume gives a detailed account of the most recent research in the area. The lectures cover a wide range of topics such as general isometric group actions, circle actions on positively curved four manifolds, cohomogeneity one actions on Alexandrov spaces, isometric torus actions on Riemannian manifolds of maximal symmetry rank, n-Sasakian manifolds, isoparametric hypersurfaces in spheres, contact CR and CR submanifolds, Riemannian submersions and the Hopf conjecture with symmetry. Also included is an introduction to the theory of exterior differential systems.

This book is not an intellectual history or popular summary of recent work on consciousness in humans. Bernard Baars (1988), Edelman and Tononi (2000), and many others, have written such, and done it well indeed. This book, rather, brings the powerful analytic machinery of communication theory to bear on the Global Neuronal Workspace (GNW) model of consciousness which Baars introduced, and does so in a formal mathematical manner. It is not the first such attempt. The philospher Fred Dretske (1981), indep- dent of Baars, long ago outlined how information theory might illuminate the understanding of mind. Adapting his approach on the necessary conditions for mental process, we apply a previously-developed information theory analysis of interacting cognitive biological and social modules to Baars' GNW, which has become the principal candidate for a 'standard model' of consciousness. Invoking an obvious canonical homology with statistical physics, the method, when iterated in the spirit of the Hierarchical Linear Model of regression theory, generates a fluctuating dynamic threshold for consciousness which is similar to a tunable phase transition in a physical system. The phenomenon is, however, constrained to a manifold/atlas structure analogous to a retina; an adaptable Rate Distortion manifold, whose 'topology', in a large sense, reflects the hierarchy of embedding constraints acting on consciousness. This view greatly extends what Baars has characterized as 'contexts.

In many areas of mathematics some "higher operations" are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.

The theory of function spaces endowed with the topology of point wise convergence, or Cp-theory, exists at the intersection of three important areas of mathematics: topological algebra, functional analysis, and general topology. Cp-theory has an important role in the classification and unification of heterogeneous results from each of these areas of research. Through over 500 carefully selected problems and exercises, this volume provides a self-contained introduction to Cp-theory and general topology. By systematically introducing each of the major topics in Cp-theory, this volume is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research. Key features include: - A unique problem-based introduction to the theory of function spaces. - Detailed solutions to each of the presented problems and exercises. - A comprehensive bibliography reflecting the state-of-the-art in modern Cp-theory. - Numerous open problems and directions for further research. This volume can be used as a textbook for courses in both Cp-theory and general topology as well as a reference guide for specialists studying Cp-theory and related topics. This book also provides numerous topics for PhD specialization as well as a large variety of material suitable for graduate research.

This volume is the collection of papers dedicated to Yozo Matsushima on his 60th birthday, which took place on February 11, 1980. A conference in Geometry in honor of Professor Matsushima was held at the University of Notre Dame on May 14 and 15, 1980. Some of the papers in this volume were delivered on this occasion. 0 00 0\ - 15 S. Kobayashi, University 27 R. Ogawa, Loyola 42 P. Ryan, Indiana 1 W. Stoll 2 W. Kaup, University of of California at Berkeley University (Chicago) University at South Bend Tubing en 16 B. Y. Chen, 28 A. Howard 43 M. Kuga, SUNY at 3 G. Shimura, Michigan State University 29 D. Blair, Stony Brook Princeton University 17 G. Ludden, Michigan State University 44 W. Higgins 30 B. Smyth 4 A. Borel, Institute for Michigan State University 45 J. Curry Advanced Study 18 S. Harris, 31 A. Pradhan 46 D. Norris 32 R. Escobales, 5 Y. Matsushima University of Missouri 47 J. Spellecy Canisius College 6 Mrs. Matsushima 19 J. Beem, 48 M. Clancy 7 K. Nomizu, University of Missouri 33 L. Smiley 49 J. Rabinowitz, University 20 D. Collins, 34 C. H. Sung Brown University of Illinois at Chicago Valparaiso University 35 M. Markowitz 8 J. -1. Hano, 50 R. Richardson, Australian Washington University 36 A. Sommese 21 I. Satake, University of National University California at Berkeley 37 A. Vitter, 9 J. Carrell, University of 51 D. Lieberman, 22 H.

The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and remains a useful and powerful tool in such areas as Lie theory, representation theory, integrable systems, complex geometry, and mathematical physics. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits (1962), places him as the founder of orbit theory. The original research papers in this volume are written by prominent mathematicians and reflect recent achievements in orbit theory and other closely related areas such as harmonic analysis, classical representation theory, Lie superalgebras, Poisson geometry, and quantization. Contributors: A. Alekseev, J. Alev, V. Baranovksy, R. Brylinski, J. Dixmier, S. Evens, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, P.W. Michor, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses.

Motility is a fundamental property of living systems, from the cytoplasmic streaming of unicellular organisms to the most highly differentiated and devel oped contractile system of higher organisms, striated muscle. Research on var ious aspects of motile systems in muscle and undifferentiated or non muscle cells has been developing at an ever more rapid pace in the laboratories of investiga tors with a wide variety of backgrounds using methodologies varying from me chanics to the most sophisticated physical measurements. Significant contri butions to our understanding of motility are coming from the disciplines of cell biology, biochemistry, pharmacology, molecular biology, biophysics, and physiology. The findings have relevance not only to basic scientists but to clinicians in such diverse fields as cardiology and neurology and to scientists in the more traditional physical sciences. Cell and Muscle Motility is a new multivolume series of essays by distinguished research workers in various fields whose work has a common thread of dealing with one aspect or another of motility. The essays are meant to focus on topics of current interest, to be critical rather than exhaustive, and to indicate the current trends of research efforts. The series is intended to foster an interchange of concepts among various workers in the field and to serve as a reference for students and workers who wish to familiarize themselves with the most current progress in motility.

The present volume grew out of the Heidelberg Knot Theory Semester, organized by the editors in winter 2008/09 at Heidelberg University. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical physics and cell biology. Both original research and survey articles are presented; numerous illustrations support the text. The book will be of great interest to researchers in topology, geometry, and mathematical physics, graduate students specializing in knot theory, and cell biologists interested in the topology of DNA strands.

Combinatorics as a branch of mathematics studies the arts of counting. Enumeration occupies the foundation of combinatorics with a large range of applications not only in mathematics itself but also in many other disciplines. It is too broad a task to write a book to show the deep development in every corner from this aspect. This monograph is intended to provide a unified theory for those related to the enumeration of maps. For enumerating maps the first thing we have to know is the sym metry of a map. Or in other words, we have to know its automorphism group. In general, this is an interesting, complicated, and difficult problem. In order to do this, the first problem we meet is how to make a map considered without symmetry. Since the beginning of sixties when Tutte found a way of rooting on a map, the problem has been solved. This forms the basis of the enumerative theory of maps. As soon as the problem without considering the symmetry is solved for one kind of map, the general problem with symmetry can always, in principle, be solved from what we have known about the automorphism of a polyhedron, a synonym for a map, which can be determined efficiently according to another monograph of the present author Liu58]."

In this book we display the fundamental structure underlying classical electro dynamics, i. e., the phenomenological theory of electric and magnetic effects. The book can be used as a textbook for an advanced course in theoretical electrodynamics for physics and mathematics students and, perhaps, for some highly motivated electrical engineering students. We expect from our readers that they know elementary electrodynamics in the conventional (1 + 3)-dimensional form including Maxwell's equations. More over, they should be familiar with linear algebra and elementary analysis, in cluding vector analysis. Some knowledge of differential geometry would help. Our approach rests on the metric-free integral formulation of the conservation laws of electrodynamics in the tradition of F. Kottler (1922), E. Cartan (1923), and D. van Dantzig (1934), and we stress, in particular, the axiomatic point of view. In this manner we are led to an understanding of why the Maxwell equa tions have their specific form. We hope that our book can be seen in the classical tradition of the book by E. J. Post (1962) on the Formal Structure of Electro magnetics and of the chapter "Charge and Magnetic Flux" of the encyclopedia article on classical field theories by C. Truesdell and R. A. Toupin (1960), in cluding R. A. Toupin's Bressanone lectures (1965); for the exact references see the end of the introduction on page 11. ."

Boundary problems constitute an essential field of common mathematical interest. The intention of this volume is to highlight several analytic and geometric aspects of boundary problems with special emphasis on their interplay. It includes surveys on classical topics presented from a modern perspective as well as reports on current research.

The collection splits into two related groups:

- analysis and geometry of geometric operators and their index theory

- elliptic theory of boundary value problems and the Shapiro-Lopatinsky condition

From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." -Zentralblatt Math

"This monograph contains a wealth of information many topologists will find very handy. ...Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature." -Mathematical Reviews

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas. Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preua * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina

In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e., characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation)."

In succesion to former international meetings on differential geometry held in Hungary and also as a satellite conference of ECM96, the European Mathematical Congress, a Conference on Differential Geometry took place in Budapest from July 27 to July 30, 1996. The host of the Conference was Lorand Eotvos University. The Conference had the following Programme Committee: D.V. Alekseevsky, J.J. Duistermaat, J. Eells, A. Haefliger, O. Kowalski, S. Marchifava, J. Szenthe, L. Tamassy, L. Vanhecke. The participants came mainly from Europe and their total number was 190. The programme included plenary lectures by J. Eliashberg, S. Gallot, O. Kowalski, B. Leeb, and also 135 lectures in 4 sections. The social events, an opening reception and a farewel party, presented inspiring atmosphere to create scientific contacts and also for fruitful discussions. In preparation of the Conference and during it B. Csikos and G. Moussong were constanly ready to help. The present volume contains detailed versions of lectures presented at the Conference and also a list of participants. The subjects cover a wide variety of topics in differential geometry and its applications and all of them contain essential new developments in their respective subjects. It is my pleasant duty to thank the participants who contributed to the success of the Conference, especially those who offered us their manuscripts for publication and also the referees who made several important observa- tions. The preparation of the volume was managed with the assistance of E. Daroczy-Kiss.

In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation, as well as for the existence and persistence of fibrations of these invariant manifolds. Their techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Fenichels results. As such, they may be applied to a broad class of infinite dimensional dynamical systems.

Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor- mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func- tions: the volume functions, which appear in the Archimedes-Newton problem on in- tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada- mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).