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.

'The book is an engaging and influential collection of significant contributions from an assembly of world expert leaders and pioneers from different fields, working at the interface between topology and physics or applications of topology to physical systems ... The book explores many interesting and novel topics that lie at the intersection between gravity, quantum fields, condensed matter, physical cosmology and topology ... A rich, well-organized, and comprehensive overview of remarkable and insightful connections between physics and topology is here made available to the physics reader.'Contemporary PhysicsSince its birth in Poincare's seminal 1894 'Analysis Situs', topology has become a cornerstone of mathematics. As with all beautiful mathematical concepts, topology inevitably - resonating with that Wignerian principle of the effectiveness of mathematics in the natural sciences - finds its prominent role in physics. From Chern-Simons theory to topological quantum field theory, from knot invariants to Calabi-Yau compactification in string theory, from spacetime topology in cosmology to the recent Nobel Prize winning work on topological insulators, the interactions between topology and physics have been a triumph over the past few decades.In this eponymous volume, we are honoured to have contributions from an assembly of grand masters of the field, guiding us with their world-renowned expertise on the subject of the interplay between 'Topology' and 'Physics'. Beginning with a preface by Chen Ning Yang on his recollections of the early days, we proceed to a novel view of nuclei from the perspective of complex geometry by Sir Michael Atiyah and Nick Manton, followed by an entree toward recent developments in two-dimensional gravity and intersection theory on the moduli space of Riemann surfaces by Robbert Dijkgraaf and Edward Witten; a study of Majorana fermions and relations to the Braid group by Louis H Kauffman; a pioneering investigation on arithmetic gauge theory by Minhyong Kim; an anecdote-enriched review of singularity theorems in black-hole physics by Sir Roger Penrose; an adventure beyond anyons by Zhenghan Wang; an apercu on topological insulators from first-principle calculations by Haijun Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum information theory as one of the four revolutions in physics and the second quantum revolution by Xiao-Gang Wen. We hope that this book will serve to inspire the research community.

Describes orthgonal and related Lie groups, using real or complex parameters and indefinite metrics. Develops theory of spinors by giving a purely geometric definition of these mathematical entities. Covers generalities on the group of rotations in n-dimensional space, the theory of spinors in spaces of any number of dimensions and much more.

This book collects select papers presented at the International Workshop and Conference on Topology & Applications, held in Kochi, India, from 9-11 December 2018. The book discusses topics on topological dynamical systems and topological data analysis. Topics are ranging from general topology, algebraic topology, differential topology, fuzzy topology, topological dynamical systems, topological groups, linear dynamics, dynamics of operator network topology, iterated function systems and applications of topology. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world. The book is a valuable resource for researchers, scientists and engineers from both academia and industry.

UEbung macht den Meister - so ist es auch in der Mathematik. Dieses Buch enthalt rund 450 Aufgaben aus verschiedenen Themenbereichen der Analysis II, die der Leserin/dem Leser dieses Buches beim Selbststudium, der hauslichen Nacharbeit des Vorlesungsstoffes und der Klausurvorbereitung helfen sollen. Dabei ist das Buch als ein Begleitwerkzeug zu verstehen, das die eifrige Leserin/den eifrigen Leser beim eigenstandigen Entwickeln von Loesungen durch gezielte Hinweise und verstandliche Loesungen unterstutzen soll. Sollten bei der Bearbeitung der Aufgaben Probleme oder Fragen aufkommen, so kann der entsprechende Loesungshinweis im zweiten Teil des Buches nachgeschlagen werden. Die eigens entwickelte Loesung der Leserin/des Lesers kann dann im Teil Loesungen mit der detaillierten und verstandlich geschriebenen Loesung abgeglichen werden. Der letzte Teil dieses Buches enthalt funf UEbungsklausuren mit unterschiedlichem Umfang, Schwierigkeitsgrad und Fokus auf einzelne Resultate und Methoden aus der Analysis II, mit denen sich die Leserin/der Leser auf eine schriftliche Prufung vorbereiten kann. Da die Vorlesung Analysis II von Universitat zu Universitat mit teilweise sehr unterschiedlichen Schwerpunkten gehalten wird, ist es denkbar, dass einige Themenbereich, die in diesem Buch behandelt werden, eher in die Analysis III oder in ein anderes Fach eingeordnet werden koennen. Dieses Buch koennte damit also auch fur Leserinnen/Leser von Interesse sein, die gerade die Vorlesung Vektoranalysis, Mass- und Integrationstheorie, Funktionalanalysis oder gewoehnliche Differentialgleichungen besuchen.

A New World of Geometry Awaits Your Discovery! The last stone falls out ... a rush of ancient air ... the glint of gold ... the tingle of discovery ... When explorers first opened the tombs of the ancient pharaohs, they knew that they had discovered something wonderful. That feeling, that same passionate sense of discovery, is one of the most powerful educational tools a text can deliver. Geometry by Discovery is an exciting new approach to geometry. This ground-breaking text taps the pedagogical value of discovery to help students stretch their geometric perspective and hone their geometric intuition. It actively engages students in solving mathematical problems, and empowers them to be successful problem-solvers and discoverers of mathematical ideas.

Inspiring popular video games like Tetris while contributing to the study of combinatorial geometry and tiling theory, polyominoes have continued to spark interest ever since their inventor, Solomon Golomb, introduced them to puzzle enthusiasts several decades ago. In this fully revised and expanded edition of his landmark book, the author takes a new generation of readers on a mathematical journey into the world of the deceptively simple polyomino. Golomb incorporates important, recent developments, and poses problems, inviting the reader to play with and develop an understanding of the extraordinary properties of polyominoes.

This research-level book presents up-to-date information concerning recent developments in convex functions and partial orderings and some applications in mathematics, statistics, and reliability theory. The book will serve researchers in mathematical and statistical theory and theoretical and applied reliabilists.
Key Features
* Presents classical and newly published results on convex functions and related inequalities
* Explains partial ordering based on arrangement and their applications in mathematics, probability, statsitics, and reliability
* Demonstrates the connection of partial ordering with other well-known orderings such as majorization and Schur functions
* Will generate further research and applications

This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises.

This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface. Isroil Ikromov and Detlef Muller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Muller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger. Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.

Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries.

Dieses Buch ist ein wichtiges studienbegleitendes Hilfsmittel fur alle, die Mathematik-Lehrveranstaltungen besuchen. Die Lekture dieses Buch ermoeglicht Ihnen, begriffliche Sicherheit fur Mathematik-Vorlesungen und Prufungen aufzubauen. Die Lekture jedes Kapitel dieses Buches erlaubt Ihnen, einen UEberblick uber die Begriffe eines Teilgebiets der Mathematik zu erhalten und diese Begriffe nachhaltig zu erfassen. Wenn Sie als Student einen mathematischen Begriff nicht richtig verstehen oder sich an seine Definition nicht erinnern, koennen Sie in diesem Buch nachschlagen und erhalten durch paradigmatische Beispiele und Bilder ein fundiertes Verstandnis des Begriffs. Arbeiten Sie ein Kapitel dieses Buches in Vorbereitung einer Prufung durch, so koennen Sie sich in begrifflicher Hinsicht in der Prufung sicher fuhlen. Insgesamt finden sich in diesem Buch mehr als tausend Definitionen von Begriffen aus vierzehn Teilgebieten der Mathematik. Die Auswahl der Begriffe orientiert sich in jedem Kapitel an den Vorlesungen zum behandelten Thema, die an deutschen Hochschulen gehalten werden. Alle wesentlichen Begriffe, die in Mathematik-Vorlesungen in Bachelorstudiengangen vorkommen und auch alle grundlegenden Begriffe der Mathematik-Vorlesungen in Masterstudiengangen sind in diesem Buch enthalten. Dieses Buch stellt also einen Kanon mathematischer Begriffe vor, der auch fur Lehrende von Interesse ist. Die 2. Auflage ist vollstandig durchgesehen und um acht neue Abschnitte zu weiterfuhrenden Themen wie etwa Simplizialkomplexen und Homologiegruppen sowie Differenzialformen erweitert.

Like Descartes and Pascal, Hans Hahn (1879-1934) was both an eminent mathematician and a highly influential philosopher. He founded the Vienna Circle and was the teacher of both Kurt Goedel and Karl Popper. His seminal contributions to functional analysis and general topology had a huge impact on the development of modern analysis. Hahn's passionate interest in the foundations of mathematics, vividly described in Sir Karl Popper's foreword (which became his last essay), had a decisive influence upon Goedel. Like Freud, Musil and Schoenberg, Hahn became a pivotal figure in the feverish intellectual climate of Vienna between the two wars. Volume 1: The first volume of Hahn's Collected Works contains his path-breaking contributions to functional analysis, the theory of curves, and ordered groups. These papers are commented on by Harro Heuser, Hans Sagan, and Laszlo Fuchs. Volume 2: The second volume deals with functional analysis, real analysis and hydrodynamics. The commentaries are written by Wilhelm Frank, Davis Preiss, and Alfred Kluwick. Volume 3: In the third volume, Hahn's writings on harmonic analysis, measure and integration, complex analysis and philosophy are collected and commented on by Jean-Pierre Kahane, Heinz Bauer, Ludger Kaup, and Christian Thiel. This volume also contains excerpts of Hahn's letters and accounts by his students and colleagues.

Like Descartes and Pascal, Hans Hahn (1879-1934) was both an eminent mathematician and a highly influential philosopher. He founded the Vienna Circle and was the teacher of both Kurt Goedel and Karl Popper. His seminal contributions to functional analysis and general topology had a huge impact on the development of modern analysis. Hahn's passionate interest in the foundations of mathematics, vividly described in Sir Karl Popper's foreword (which became his last essay), had a decisive influence upon Goedel. Like Freud, Musil and Schoenberg, Hahn became a pivotal figure in the feverish intellectual climate of Vienna between the two wars. Volume 1: The first volume of Hahn's Collected Works contains his path-breaking contributions to functional analysis, the theory of curves, and ordered groups. These papers are commented on by Harro Heuser, Hans Sagan, and Laszlo Fuchs. Volume 2: The second volume deals with functional analysis, real analysis and hydrodynamics. The commentaries are written by Wilhelm Frank, Davis Preiss, and Alfred Kluwick. Volume 3: In the third volume, Hahn's writings on harmonic analysis, measure and integration, complex analysis and philosophy are collected and commented on by Jean-Pierre Kahane, Heinz Bauer, Ludger Kaup, and Christian Thiel. This volume also contains excerpts of Hahn's letters and accounts by his students and colleagues.

Aus den Rezensionen: "Was das Buch vor allem auszeichnet, ist die unkonventionelle Darstellungsweise. Hier wird Mathematik nicht im trockenen Definition-Satz-Beweis-Stil geboten, sondern sie wird dem Leser pointiert und mit viel Humor schmackhaft gemacht. In ungew hnlich fesselnder Sprache geschrieben, ist die Lekt re dieses Buches auch ein belletristisches Vergn gen. Fast 200 sehr instruktive und sch ne Zeichnungen unterst tzen das Verst ndnis, motivieren die behandelten Aussagen, modellieren die tragenden Beweisideen heraus. Ungew hnlich ist auch das Register, das unter jedem Stichwort eine Kurzdefinition enth lt und somit umst ndliches Nachschlagen erspart."
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Jetzt in der achten Auflage des bew hrten Lehrbuches!

Nato dall'esperienza dell'autore nell'insegnamento della topologia agli studenti del corso di Laurea in Matematica, questo libro contiene le nozioni fondamentali di topologia generale ed una introduzione alla topologia algebrica. La scelta degli argomenti, il loro ordine di presentazione e, soprattutto, il tipo di esposizione tiene conto delle tendenze attuali nell'insegnamento della topologia e delle novita nella struttura dei corsi di Laurea scientifici conseguenti all'introduzione del sistema 3+2. Questa seconda edizione, oltre a semplificare alcune dimostrazioni, presenta una sostanziale riscrittura della parte sui rivestimenti e l'aggiunta di ulteriori esempi; il numero complessivo di esercizi proposti stato portato a 500 ed il numero di quelli svolti a 120.

Wie bewegt sich ein Massenpunkt in einem Gebiet, an dessen Rand er elastisch zuruckprallt? Welchen Weg nimmt ein Lichtstrahl in einem Gebiet mit ideal reflektierenden Randern? Anhand dieser und ahnlicher Fragen stellt das vorliegende Buch Zusammenhange zwischen Billard und Differentialgeometrie, klassischer Mechanik sowie geometrischer Optik her. Dabei beschaftigt sich das Buch unter anderem mit dem Variationsprinzip beim mathematischen Billard, der symplektischen Geometrie von Lichtstrahlen, der Existenz oder Nichtexistenz von Kaustiken, periodischen Billardtrajektorien und dem Mechanismus fur Chaos bei der Billarddynamik. Erganzend wartet dieses Buch mit einer beachtlichen Anzahl von Exkursen auf, die sich verwandten Themen widmen, darunter der Vierfarbensatz, die mathematisch-physikalische Beschreibung von Regenbogen, der poincaresche Wiederkehrsatz, Hilberts viertes Problem oder der Schliessungssatz von Poncelet.

Wahrend einer Konferenz zum "Jiidischen Nietzscheanismus" 1995 in Greifs wald hatte mich EGBERT BRIESKORN eingeladen, in der Edition der Gesam melten Werke FELIX HAUSDORFFS dessen philosophische Schriften mit einer Einleitung herauszugeben. FELIX HAUSDORFF hatte darin eng an NIETZSCHE angeschlossen, und er hatte in Greifswald sein erstes Ordinariat fUr Mathematik erhalten - ich sagte spontan und, wie sich bald herausstellen soUte, leichtsinnig ja. Statt nur mit einer kurzen Einleitung hatte ich es bald auch mit langwieri gen Erschlief&ungen des Werks und seiner Kommentierung zu tun. Doch je mehr ich mich in FELIX HAUSDORFFS Schriften einarbeitete, desto mehr notigten sie mir Respekt ab: in ihrer Klarheit, ihrer Redlichkeit, ihrer vornehmen Beschei denheit, ihrer gedanklichen Selbstandigkeit und vor allem in ihrer erstaunlichen Aktualitat. Vielleicht ist nach iiber hundert Jahren nun die Zeit gekommen, in der sie fiir die philosophische Orientierung so fruchtbar werden konnen, wie sie es verdienen. Bei der Kommentierung haben viele helfende Hande mitgewirkt. Mein Dank gilt zuerst den studentischen und wissenschaftlichen Hilfskraften: MIRKO GRON DER und KATRIN STELTER haben die Hauptarbeit in der Recherchierung der Belege iibernommen, JUDITH KARLA und TANJA SCHMIDT eine Vielzahl von Nachweisen beigesteuert, WOLFGANG SCHNEIDER und RALF WITZLER an den Vorarbeiten mitgewirkt. Doz. Dr. REINHARD PESTER (friiher Greifswald, jetzt Berlin) hat uns bei den Nachweisen zu LOTZE, Prof. Dr. MARTIN HOSE (frii her Greifswald, jetzt Miinchen) bei Zitaten aus der griechischen Literatur, Prof. Dr. GISELA FEBEL (friiher Stuttgart, jetzt Bremen) bei Zitaten aus der franzosischen Literatur, Prof. Dr. WALTER ERHART, Prof. Dr."

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.

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.

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.

In this book, I. Ya. Bakel'man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the point of infinity and the stereographic projection of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane. The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy's theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry."