The aim of this book is to study harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. In several instances, the latter permits reduction of the original elliptic variational problem to the qualitative study of certain ordinary differential equations: the authors' primary objective is to provide representative examples to illustrate these reduction methods and their associated analysis with geometric and topological applications.

The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.

After the development of manifolds and algebraic varieties in the previous century, mathematicians and physicists have continued to advance concepts of space. This book and its companion explore various new notions of space, including both formal and conceptual points of view, as presented by leading experts at the New Spaces in Mathematics and Physics workshop held at the Institut Henri Poincare in 2015. The chapters in this volume cover a broad range of topics in mathematics, including diffeologies, synthetic differential geometry, microlocal analysis, topos theory, infinity-groupoids, homotopy type theory, category-theoretic methods in geometry, stacks, derived geometry, and noncommutative geometry. It is addressed primarily to mathematicians and mathematical physicists, but also to historians and philosophers of these disciplines.

The ideal review for your general topology course

More than 40 million students have trusted Schaum's Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaum's Outlines cover everything from math to science, nursing to language. The main feature for all these books is the solved problems. Step-by-step, authors walk readers through coming up with solutions to exercises in their topic of choice. 391 solved problems 356 supplementary problems Teaches effective problem-solving Outline format supplies a concise guide to the standard college courses in General Topology Supports and supplements the leading General Topology textbooks Detailed explanations and practice problems in general topology Comprehensive review of specialized topics in topology

This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.

"Three-Dimensional Geometry and Topology" had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincare Conjecture.

In 2005 Thurston won the first AMS Book Prize, for "Three-dimensional Geometry and Topology." The prize recognizes an outstanding research book that makes a seminal contribution to the research literature. Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation."

The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatial geometries, and one scalar function of time. Combining the cosmological principle with an appropriate description of the matter leads to the standard models. It is worth noting that these models yield quite a successful description of our universe. However, even though the universe may, or may not, be almost spatially homogeneous and isotropic, it is clear that the cosmological principle is not exactly satisfied. This leads to several questions. The most natural one concerns stability: given initial data corresponding to an expanding model of the standard type, do small perturbations give rise to solutions that are similar to the future? Another question concerns the shape of the universe: what are the restrictions if we only assume the universe to appear almost spatially homogeneous and isotropic to every observer? The main purpose of the book is to address these questions. However, to begin with, it is necessary to develop the general theory of the Cauchy problem for the Einstein-Vlasov equations. In order to to make the results accessible to researchers who are not mathematicians, but who are familiar with general relativity, the book contains an extensive prologue putting the results into a more general context.

Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago.

The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory.

The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

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.

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.

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.""

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.

Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle.

Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

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.

This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and Andre Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincare-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.

In this book Francois De Gandt introduces us to the reading of Newton's Principia in its own terms. The path of access that De Gandt proposes leads through the study of the geometrization of force. The result is a highly original meditation on the sources and meaning of Newton's magnum opus. In Chapter I De Gandt presents a translation of and detailed commentary on an earlier and simpler version of what in 1687 became Book I of the Principia; here in clearer and starker outline than in the final version, the basic principles of Newton's dynamics show forth. Chapter II places this dynamics in the intellectual context of earlier efforts--the first seeds of celestial dynamics in Kepler, Galileo's theory of accelerated motion, and Huygens's quantification of centrifugal force--and evaluates Newton's debt to these thinkers. Chapter III is a study of the mathematical tools used by Newton and their intellectual antecedents in the works of Galileo, Torricelli, Barrow, and other seventeenth-century mathematicians. The conclusion discusses the new status of force and cause in the science that emerges from Newton's Principia. Originally published in 1995. 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.

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.

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.