Topics in Geometric Group Theory


Author: Pierre de la Harpe
Publisher: University of Chicago Press
ISBN: 9780226317199
Category: Mathematics
Page: 310
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In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, 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 adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing 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." 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 problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields.

Geometrische Gruppentheorie

Ein Einstieg mit dem Computer Basiswissen für Studium und Mathematikunterricht
Author: Stephan Rosebrock
Publisher: Springer-Verlag
ISBN: 9783834810380
Category: Mathematics
Page: 211
View: 6873

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In diesem Buch geht es um Gruppentheorie. Man kann Gruppen als algebraische Objekte auffassen, die die Symmetrie von geometrischen Objekten beschreiben. Dieser Blickwinkel steht bei dem Buch im Vordergrund und somit geht es in dem Buch auch um Geometrie. Gruppen drücken Symmetriephänomene algebraisch aus, man rechnet mit Spiegelungen, Drehungen usw., allgemein mit Abbildungen von Räumen auf sich. Das Buch kann vorlesungsbegleitend bei Algebra- und Gruppentheorie-Vorlesungen eingesetzt werden. Es eignet sich auch besonders gut für Lehramtsstudierende, da es den Stoff computerorientiert (unter Benutzung des frei erhältlichen Gruppentheorie-Programms GAP) mit vielen anschaulichen Beispielen präsentiert. Für die 2. Auflage wurden einige Teile des Buches ausführlicher dargestellt. Einige Inhalte (zum Beispiel die Klassifikation der endlichen Gruppen bis zur Ordnung 11) wurden hinzugefügt und Fehler korrigiert. Einführung in die euklidische Geometrie – Einführung in Gruppen – Untergruppen und Homomorphismen – Gruppenoperationen – Gruppenpräsentationen – Produkte von Gruppen – Endliche Gruppen – Die hyperbolische Ebene – Hyperbolische Gruppen - Studierende der Mathematik und Naturwissenschaften ab dem 3. Semester - Studierende der Lehrämter - Mathematiklehrerinnen und -lehrer - Dozenten der Mathematik und ihrer Didaktik Akad. Oberrat Dr. Stephan Rosebrock ist Dozent für Mathematik an der PH Karlsruhe, Institut für Mathematik und Informatik.

Computational and Statistical Group Theory

AMS Special Session Geometric Group Theory, April 21-22, 2001, Las Vegas, Nevada, AMS Special Session Computational Group Theory, April 28-29, 2001, Hoboken, New Jersey
Author: Esther G Forbes,Robert H. Gilman,Alexei G. Myasnikov,Vladimir Shpilrain,Nev.) AMS Special Session Geometric Group Theory (2001 : Las Vegas
Publisher: American Mathematical Soc.
ISBN: 0821831585
Category: Mathematics
Page: 124
View: 9244

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This book gives a nice overview of the diversity of current trends in computational and statistical group theory. It presents the latest research and a number of specific topics, such as growth, black box groups, measures on groups, product replacement algorithms, quantum automata, and more. It includes contributions by speakers at AMS Special Sessions at The University of Nevada (Las Vegas) and the Stevens Institute of Technology (Hoboken, NJ). It is suitable for graduate students and research mathematicians interested in group theory.

Groups, Geometry and Dynamics


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Group theory
Page: N.A
View: 7399

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Publishes research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. Covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.

Lectures on Exceptional Lie Groups


Author: J. F. Adams
Publisher: University of Chicago Press
ISBN: 9780226005270
Category: Mathematics
Page: 122
View: 4042

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J. Frank Adams was internationally known and respected as one of the great algebraic topologists. Adams had long been fascinated with exceptional Lie groups, about which he published several papers, and he gave a series of lectures on the topic. The author's detailed lecture notes have enabled volume editors Zafer Mahmud and Mamoru Mimura to preserve the substance and character of Adams's work. Because Lie groups form a staple of most mathematics graduate students' diets, this work on exceptional Lie groups should appeal to many of them, as well as to researchers of algebraic geometry and topology. J. Frank Adams was Lowndean professor of astronomy and geometry at the University of Cambridge. The University of Chicago Press published his Lectures on Lie Groups and has reprinted his Stable Homotopy and Generalized Homology. Chicago Lectures in Mathematics Series

Fuchsian Groups


Author: Svetlana Katok
Publisher: University of Chicago Press
ISBN: 9780226425825
Category: Mathematics
Page: 175
View: 625

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This introductory text provides a thoroughly modern treatment of Fuchsian groups that addresses both the classical material and recent developments in the field. A basic example of lattices in semisimple groups, Fuchsian groups have extensive connections to the theory of a single complex variable, number theory, algebraic and differential geometry, topology, Lie theory, representation theory, and group theory.

Geometry, Rigidity, and Group Actions


Author: Robert J Zimmer,Benson Farb,David Fisher
Publisher: University of Chicago Press
ISBN: 0226237893
Category: Mathematics
Page: 646
View: 5794

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The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others. The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.

A Concise Course in Algebraic Topology


Author: J. P. May
Publisher: University of Chicago Press
ISBN: 9780226511832
Category: Mathematics
Page: 243
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Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.

Klassische Elektrodynamik


Author: John David Jackson
Publisher: Walter de Gruyter
ISBN: 311033447X
Category: Science
Page: 957
View: 3595

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„In der gesamten physikalischen Lehrbuchliteratur gibt es wohl kaum ein anderes Werk, das auf seinem Feld so unangefochten eine Spitzenstellung behauptet wie das Elektrodynamik-Buch von Jackson, und das bereits seit vier Jahrzehnten." – Physik Journal. Die deutsche Übersetzung dieses Klassikers der theoretischen Physik erscheint jetzt in einer nochmals durchgesehenen Neuauflage.Damit wirdtheoretische Elektrodynamik noch verständlicher als je zuvor. Einzigartig bleibt die konkurrenzlos hohe Anzahl von konkret gerechneten Beispielen, exakt durchgerechneten Fällen und zahlreichen Übungsaufgaben. Nach wie vor ist das Buch seit der 3. Auflage größtenteils in SI geschrieben. Seine Anwendungsnähe (auch zur Experimentalphysik) wird sowohl von Studenten als auch von Wissenschaftlern, Hochschullehrern und Ingenieuren geschätzt.

Lectures on Mathematics


Author: Felix Klein
Publisher: American Mathematical Soc.
ISBN: 0821827332
Category: Mathematics
Page: 109
View: 4349

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In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and reissued by the AMS in 1911, we are pleased to bring this work into print once more with this new edition. Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry. Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron. Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about ``physical mathematics''. There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need for a theory of ideals as developed by Kummer. Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era.

Simplicial Objects in Algebraic Topology


Author: J. P. May
Publisher: University of Chicago Press
ISBN: 9780226511818
Category: Mathematics
Page: 161
View: 4072

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Simplicial sets are discrete analogs of topological spaces. They have played a central role in algebraic topology ever since their introduction in the late 1940s, and they also play an important role in other areas such as geometric topology and algebraic geometry. On a formal level, the homotopy theory of simplicial sets is equivalent to the homotopy theory of topological spaces. In view of this equivalence, one can apply discrete, algebraic techniques to perform basic topological constructions. These techniques are particularly appropriate in the theory of localization and completion of topological spaces, which was developed in the early 1970s. Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets, together with the equivalence of homotopy theories alluded to above. The central theme is the simplicial approach to the theory of fibrations and bundles, and especially the algebraization of fibration and bundle theory in terms of "twisted Cartesian products." The Serre spectral sequence is described in terms of this algebraization. Other topics treated in detail include Eilenberg-MacLane complexes, Postnikov systems, simplicial groups, classifying complexes, simplicial Abelian groups, and acyclic models. "Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint. It should prove very valuable to anyone wishing to learn semi-simplicial topology. [May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."—Mathematical Review