This book is a remarkable contribution to the literature on dynamical systems and geometry. It consists of a selection of work in current research on Teichmuller dynamics, a field that has continued to develop rapidly in the past decades. After a comprehensive introduction, the author investigates the dynamics of the Teichmuller flow, presenting several self-contained chapters, each addressing a different aspect on the subject. The author includes innovative expositions, all the while solving open problems, constructing examples, and supplementing with illustrations. This book is a rare find in the field with its guidance and support for readers through the complex content of moduli spaces and Teichmuller Theory. The author is an internationally recognized expert in dynamical systems with a talent to explain topics that is rarely found in the field. He has created a text that would benefit specialists in, not only dynamical systems and geometry, but also Lie theory and number theory.

This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

The aim of this handbook is to create, for the first time, a systematic account of the field of spatial logic. The book comprises a general introduction, followed by fourteen chapters by invited authors. Each chapter provides a self-contained overview of its topic, describing the principal results obtained to date, explaining the methods used to obtain them, and listing the most important open problems. Jointly, these contributions constitute a comprehensive survey of this rapidly expanding subject.

Many of the developments of modern algebraic geometry and topology stem from the ideas of S. Lefschetz. These are featured in this volume of contemporary research papers contributed by mathematical colleagues to celebrate his seventieth birthday. Originally published in 1957. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Dieses Buch bietet eine erste Einfuhrung in die mathematische Theorie der dynamischen Systeme, die fur Studierende des letzten Studienjahres des Bachelor Studiums und fur das Master Studium geeignet ist. Aufbauend auf den Grundbegriffen der Topologischen Dynamik und der Ergodentheorie in den ersten beiden Kapiteln behandelt das dritte Kapitel den fur die Ergodentheorie zentralen Begriff der Entropie, der seinen Ursprung in der statistischen Physik und der Informationstheorie hat, und der die Komplexitat eines masstheoretischen dynamischen Systems quantifiziert. Das vierte Kapitel ist ebenfalls der Entropie gewidmet, diesmal aber im Rahmen der topologischen Dynamik, bei derEntropie einen quantitativen Ausdruck fur die Verformung eines kompakten metrischen Raumes durch eine stetige Transformation darstellt. Das funfte und letzte Kapitel gibt einen kleinen Einblick in aktuelle Entwicklungen der Theorie der dynamischen Systeme mit ihren mehrparametrischen Verallgemeinerungen des klassischen Konzepts der Zeitentwicklung und den daraus entspringenden und zum Teil uberraschenden Querverbindungen zu anderen mathematischen Disziplinen.
Das in Vorlesungen erprobte Material dieses Buches kann durch eine den Interessen der Studierenden angepasste Themenauswahl wahlweise fur eine ein- oder zweisemestrige Vorlesung eingesetzt werden. "

Felix Hausdorff gehort zu den herausragenden Mathematikern der ersten Halfte des 20. Jahrhunderts. Er hinterliess einen ungewohnlich reichhaltigen Korpus wissenschaftlicher Manuskripe. Sein Gesamtwerk soll nun in 9 Banden, jeweils mit detaillierten Kommentaren, herausgegeben werden. Der vorliegende Band II enthalt Hausdorffs wohl wichtigstes Werk, die "Grundzuge der Mengenlehre" Dieses Buch gehort zu den Klassikern der mathematischen Literatur und hat auf die Entwicklung der Mathematik im 20. Jahrhundert einen bedeutenden Einfluss ausgeubt. Daher erschien es geboten, ausfuhrliche Kommentare beizufugen. In diesen Kommentaren werden vor allem die bedeutenden originellen Beitrage, die Hausdorff in den "Grundzugen" zur Topologie, allgemeinen und deskriptiven Mengenlehre geleistet hat, eingehend behandelt. Insbesondere wird versucht, Hausdorffs Leistungen in die historische Entwicklung einzuordnen und ihre jeweilige Wirkungsgeschichte zu skizzieren."

Die Knotentheorie hat sich im letzten Jahrzehnt zu einem der aktivsten Forschungsgebiete in der Mathematik entwickelt. Eine Vielzahl neuer Ergebnisse wurde gefunden, die sich nicht nur in der Topologie, sondern auch in anderen Gebieten der Mathematik und sogar in anderen Naturwissenschaften wie der Physik und der Biologie fruchtbar einsetzen liessen. Diese erstaunliche Entwicklung hat eine beachtliche Zahl von Buchveroffentlichungen zur Knotentheorie zur Folge gehabt, wobei eine historische Darstellung bislang noch nicht vorliegt. Dieses Buch schliesst diese Lucke und spannt den Bogen von Gauss bis zur heutigen Knotentheorie. Allgemein verstandliche und mathematisch anspruchsvolle Abschnitte sind klar zu unterscheiden.

Die technische Entwicklung verlangt Maschinen und Mechanismen, die schneller laufen, leichter gebaut sind und genauer, zuverlassiger und wirtschaftlicher arbeiten als ihre Vorganger. Schwingungsprobleme, auf die man frtiher in der Industriepraxis nur selten stieB, verlangen eine Lasung. Die Berechnung der in Mechanismen auftretenden dynamischen Krafte und De- formationen wird durch diese steigenden Anforderungen immer wichtiger. Das vor- liegende Buch hat das Ziel, mathematische und mechanische Methoden, die zur dynamischen Analyse, Synthese und Optimierung realer Mechanismen geeignet sind, zusammenfassend darzustellen. Es wird ein Uberblick tiber solche Gebiete der Mechanismendynamik gegeben, deren Methoden und Verfahren im Maschinenbau von Interesse sind. Manchmal versuchen die Anhanger traditioneller Vorstellungen, zwischen der Theorie der Maschinen und Mechanismen bzw. der Getriebetechnik einerseits und der Angewandten Mechanik andererseits eine scharfe Grenze zu ziehen. Wir meinen, daB dafUr kein Grund besteht. Die Gebiete der Maschinendynamik, Schwingungs- lehre, Getriebelehre und Technischen Mechanik stehen auf dem gleichen Fundament der klassischen Mechanik, und mit der Weiterentwicklung der numerischen Mathe- matik und Informatik rucken sie enger zusammen.

Anschauliche Geometrie - wohl selten ist ein Mathematikbuch seinem Titel so gerecht geworden wie dieses aussergewohnliche Werk von Hilbert und Cohn-Vossen. Zuerst 1932 erschienen, hat das Buch nichts von seiner Frische und Kraft verloren. Hilbert hat sein erklartes Ziel, die Faszination der Geometrie zu vermitteln, bei Generationen von Mathematikern erreicht.

Aus Hilberts Vorwort: "Das Buch soll dazu dienen, die Freude an der Mathematik zu mehren, indem es dem Leser erleichtert, in das Wesen der Mathematik einzudringen, ohne sich einem beschwerlichen Studium zu unterziehen.""

Wahrend wir diese Zeilen niederschreiben, vollenden sich hundert Jahre seit der Entdeckung des piezoelektrischen Effektes. Seine technischen Anwendungen lieBen zwar ziemlich lange auf sich warten, sind jedoch heute kaum aus unserem Leben wegzudenken. Die piezoelektrischen Resonatoren steuern die Frequenzen von Send ern sowie den Gang von Quarzuhren, dienen als Frequenzfilter und erzeugen Ultraschallwellen. Etwas im Schatten derartiger Anwendungen machte man sich den piezoelektrischen Effekt ebenfalls zum Messen von Kraften, Drucken und Be- schleunigungen zunutze. Dieses an und fUr sich nachstliegende Anwendungsgebiet der Piezoelektrizitat wurde auch in der Literatur nur bescheiden berucksichtigt. Eine gebuhrende Aufmerksamkeit wurde ihm eigentlich nur in zwei Monogra- phien uber die Piezoelektrizitat [S 3, P 3] zuteil, wobei die erste einzig Aufnehmer mit Quarzelementen behandelt und die zweite, der Sprache wegen, nur einem be- schrankten Leserkreis zuganglich bleibt. AusschlieBlich mit der piezoelektrischen MeBtechnik beschaftigt sich das Buch von W. Gohlke [G8]. Seit dem leicht er- ganzten Nachdruck sind jedoch immerhin schon zwanzig Jahre vergangen, und die Auflage ist langst vergriffen. Die inzwischen in der deutschen wie auch in anderen Sprachen in Handbuchern der allgemeinen MeBtechnik erschienenen Darstellungen oder Firmenschriften uber spezielle Teilgebiete konnten die Lucke nicht schlieBen.

Including presentations by field authorities describing the state of current research, a workshop was held on Kleinian groups and hyperbolic 3-manifolds in September 2001. This volume includes a selection of workshop contributions representative of its extremely high standards. Beginning graduate students will find them inspiring, and established researchers will discover reliable references to current research.

This book discusses the basic geometric contents of an image and presents a treedatastructuretohandleite?ciently.Itanalyzesalsosomemorphological operators that simplify this geometric contents and their implementation in termsofthe datastructuresintroduced.It?nallyreviewsseveralapplications to image comparison and registration, to edge and corner computation, and the selection of features associated to a given scale in images. Let us ?rst say that, to avoid a long list, we shall not give references in this summary; they are obviously contained in this monograph. A gray level image is usually modeled as a function de?ned in a bounded N domain D? R (typically N = 2 for usual snapshots, N=3formedical images or movies) with values in R. The sensors of a camera or a CCD array transform the continuum of light energies to a ?nite interval of values by means of a nonlinear function g. The contrast change g depends on the pr- ertiesofthesensors,butalsoontheilluminationconditionsandthere?ection propertiesofthe objects,andthoseconditionsaregenerallyunknown.Images are thus observed modulo an arbitrary and unknown contrast change.

A concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.

Wer Mathematik liebt, findet sie spannend und aufregend. F r viele Menschen ist Mathematik allerdings ein Buch mit sieben Siegeln. Dieses Buch schl gt Br cken in das Reich der Mathematik. Es richtet sich an Leser jeden Alters und jeder Vorbildung. Gymnasiallehrer finden eine reiche Auswahl an Beispielen, Studenten bietet es Orientierung, und Dozenten werden sich an den Feinheiten der Darstellung zweier Meister ihres Faches erfreuen.

Groups as abstract structures were first recognized by mathematicians in the nineteenth century. Groups are, of course, sets given with appropriate "multiplications," and they are often given together with actions on interesting geometric objects. But groups are also interesting geometric objects by themselves. More precisely, a finitely-generated group can be seen as a metric space, the distance between two points being defined "up to quasi-isometry" by some "word length," and this gives rise to a very fruitful approach to group theory.
In this book, Pierre de la Harpe provides a concise and engaging introduction to this approach, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe uses a hands-on presentation style, illustrating key concepts of geometric group theory with numerous concrete examples.
The first five chapters present basic combinatorial and geometric group theory in a unique way, with an emphasis on finitely-generated versus finitely-presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group," an infinite finitely-generated torsion group of intermediate growth which is becoming more and more important in group theory. Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research questions in the field. An extensive list of references directs readers to more advanced results as well as connectionswith other subjects.

Originally published as Volume 27 of the Princeton Mathematical series. Originally published in 1965. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In this book, which may be used as a self-contained text for a beginning course, Professor Lefschetz aims to give the reader a concrete working knowledge of the central concepts of modern combinatorial topology: complexes, homology groups, mappings in spheres, homotopy, transformations and their fixed points, manifolds and duality theorems. Each chapter ends with a group of problems. Originally published in 1949. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In this book, which may be used as a self-contained text for a beginning course, Professor Lefschetz aims to give the reader a concrete working knowledge of the central concepts of modern combinatorial topology: complexes, homology groups, mappings in spheres, homotopy, transformations and their fixed points, manifolds and duality theorems. Each chapter ends with a group of problems. Originally published in 1949. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Starting from the foundations, the author presents an almost entirely
self-contained treatment of differentiable spaces of nonpositive
curvature, focusing on the symmetric spaces in which every geodesic lies
in a flat Euclidean space of dimension at least two. The book builds to
a discussion of the Mostow Rigidity Theorem and its generalizations, and
concludes by exploring the relationship in nonpositively curved spaces
between geometric and algebraic properties of the fundamental group.
This introduction to the geometry of symmetric spaces of non-compact
type will serve as an excellent guide for graduate students new to the
material, and will also be a useful reference text for mathematicians
already familiar with the subject.

Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of varifold surfaces and uses the methods of geometric measure theory to develop a mathematical description of the motion of a surface by its mean curvature. This mathematical description encompasses, among other subtleties, those of changing geometries and instantaneous mass losses. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

This volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.

Hauptgegenstand des Buches sind Homologie-, Kohomologietheorien und Mannigfaltigkeiten. In den ersten acht Kapiteln werden Begriffe wie Homologie, CW-Komplexe, Produkte und Poincare Dualitat eingefuhrt und deren Anwendungen diskutiert. In den davon unabhangigen Kapiteln 9 bis 13 werden Differentialformen und der Satz von Stokes auf Mannigfaltigkeiten behandelt. Die in Kapitel 14 und 15 behandelte de Rham Kohomologie und der Satz von de Rham verbinden diese beiden Teile.

Das vorliegende Buch zielt auf eine Vertiefung und Erweiterung geometrischer Kenntnisse von Studierenden der Mathematik und der Physik nach dem Grundstudium, und zwar an Hand unterschiedlicher, attraktiver, elementar-zug nglicher Themen der Geometrie. Bez ge zur Analysis und Physik werden betont, zur Historie einiger bedeutender geometrischer bzw. physikalischer Begriffe oder Fragestellungen gibt es eingehendere Beitr ge. Die generelle Ausf hrlichkeit des Textes sollte es Dozenten erm glichen, den Vortrag auf die Vermittlung der Begriffe, Resultate und Beweisideen zu konzentrieren und f r gewisse Details auf den Text verweisen zu k nnen.

The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the "cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.

During the summer of 1965, an informal seminar in geometric topology was held at the University of Wisconsin under the direction of Professor Bing. Twenty-five of these lectures are included in this study, among them Professor Bing's lecture describing the recent attacks of Haken and Poincare on the Poincare conjectures, and sketching a proof of Haken's main result.