Notes on Geometry

Author: Elmer Rees

Publisher: Springer Science & Business Media

ISBN: 9783540120537

Category: Mathematics

Page: 114

View: 8753

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In recent years, geometry has played a lesser role in undergraduate courses than it has ever done. Nevertheless, it still plays a leading role in mathematics at a higher level. Its central role in the history of mathematics has never been disputed. It is important, therefore, to introduce some geometry into university syllabuses. There are several ways of doing this, it can be incorporated into existing courses that are primarily devoted to other topics, it can be taught at a first year level or it can be taught in higher level courses devoted to differential geometry or to more classical topics. These notes are intended to fill a rather obvious gap in the literature. It treats the classical topics of Euclidean, projective and hyperbolic geometry but uses the material commonly taught to undergraduates: linear algebra, group theory, metric spaces and complex analysis. The notes are based on a course whose aim was two fold, firstly, to introduce the students to some geometry and secondly to deepen their understanding of topics that they have already met. What is required from the earlier material is a familiarity with the main ideas, specific topics that are used are usually redone.
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Topological Methods in Galois Representation Theory

Author: Victor P. Snaith

Publisher: Courier Corporation

ISBN: 0486782271

Category: Mathematics

Page: 320

View: 1366

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This advanced monograph on Galois representation theory by a renowned algebraist covers abelian and nonabelian cohomology of groups, characteristic classes of forms and algebras, explicit Brauer induction theory, more. 1989 edition.
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Algorithmic and Geometric Topics Around Free Groups and Automorphisms

Author: Javier Aramayona,Volker Diekert,Christopher J. Leininger,Pedro V. Silva,Armin Weiß

Publisher: Birkhäuser

ISBN: 3319609408

Category: Mathematics

Page: 149

View: 6718

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This volume presents the lecture notes from the authors’ three summer courses offered during the program “Automorphisms of Free Groups: Geometry, Topology, and Dynamics,” held at the Centre de Recerca Matemàtica (CRM) in Bellaterra, Spain. The first two chapters present the basic tools needed, from formal language theory (regular and context-free languages, automata, rewriting systems, transducers, etc) and emphasize their connections to group theory, mostly relating to free and virtually-free groups. The material covered is sufficient to present full proofs of many of the existing interesting characterizations of virtually-free groups. In turn, the last chapter comprehensively describes Bonahon’s construction of Thurston’s compactification of Teichmüller space in terms of geodesic currents on surfaces. It also includes several intriguing extensions of the notion of geodesic current to various other, more general settings.
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Geometry, Topology and Dynamics of Character Varieties

Author: William Goldman,Caroline Series,Ser Peow Tan

Publisher: World Scientific

ISBN: 9814401358

Category: Mathematics

Page: 349

View: 6831

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This book aims to describe, for readers uneducated in science, the development of humanity's desire to know and understand the world around us through the various stages of its development to the present, when science is almost universally recognized - at least in the Western world - as the most reliable way of knowing. The book describes the history of the large-scale exploration of the surface of the earth by sea, beginning with the Vikings and the Chinese, and of the unknown interiors of the American and African continents by foot and horseback. After the invention of the telescope, visual exploration of the surfaces of the Moon and Mars were made possible, and finally a visit to the Moon. The book then turns to our legacy from the ancient Greeks of wanting to understand rather than just know, and why the scientific way of understanding is valued. For concreteness, it relates the lives and accomplishments of six great scientists, four from the nineteenth century and two from the twentieth. Finally, the book explains how chemistry came to be seen as the most basic of the sciences, and then how physics became the most fundamental.
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Using the Borsuk-Ulam Theorem

Lectures on Topological Methods in Combinatorics and Geometry

Author: Ji?í Matou?ek,Jiri Matousek,A. Björner,G.M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 3540003622

Category: Computers

Page: 196

View: 9580

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To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.
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