The Shape of Space

Author: Jeffrey R. Weeks
Publisher: Marcel Dekker
ISBN: 082477437X
Release Date: 1985-01-01
Genre: Surfaces

The Shape of Space brings topology to the general reader by showing how to visualize manifolds directly...complements existing textbooks, which often deal only in abstractions, by offering a wealth of concrete examples...includes the first elementary exposition of William P. Thurston's revolutionary discoveries...applies topology to cosmology...gives the first simple pictorial exposition of the Gauss-Bonnet formula...builds intuition with more than 140 hands-on exercises, all with complete solutions...and offers over 170 illustrations. An annotated bibliography lists useful references for further study on specific topics.

The Shape of Space

Author: Jeffrey R. Weeks
Publisher: CRC Press
ISBN: 0824707095
Release Date: 2001-12-12
Genre: Mathematics

Maintaining the standard of excellence set by the previous edition, this textbook covers the basic geometry of two- and three-dimensional spaces Written by a master expositor, leading researcher in the field, and MacArthur Fellow, it includes experiments to determine the true shape of the universe and contains illustrated examples and engaging exercises that teach mind-expanding ideas in an intuitive and informal way. Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

The Shape of Space

Author: Jeffrey R. Weeks
Publisher: CRC Press
ISBN: 9780824748371
Release Date: 2001-12-12
Genre: Mathematics

Maintaining the standard of excellence set by the previous edition, this textbook covers the basic geometry of two- and three-dimensional spaces Written by a master expositor, leading researcher in the field, and MacArthur Fellow, it includes experiments to determine the true shape of the universe and contains illustrated examples and engaging exercises that teach mind-expanding ideas in an intuitive and informal way. Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behavior of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

Low dimensional Geometry

Author: Francis Bonahon
Publisher: American Mathematical Soc.
ISBN: 9780821848166
Release Date: 2009-07-14
Genre: Mathematics

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

The Shape of Inner Space

Author: Shing-Tung Yau
Publisher: Basic Books
ISBN: 9780465022663
Release Date: 2010-09-07
Genre: Science

String theory says we live in a ten-dimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as Calabi-Yau manifolds. In The Shape of Inner Space, Shing-Tung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe. Time and again, where Yau has gone, physics has followed. Now for the first time, readers will follow Yau’s penetrating thinking on where we’ve been, and where mathematics will take us next. A fascinating exploration of a world we are only just beginning to grasp, The Shape of Inner Space will change the way we consider the universe on both its grandest and smallest scales.

Knots and Surfaces

Author: N. D. Gilbert
Publisher: Oxford University Press, UK
ISBN: 9780191591907
Release Date: 1994-12-01
Genre: Knot theory

Completely up-to-date, illustrated throughout, and written in an accessible style, Knots and Surfaces is an account of the mathematical theory of knots and its interaction with related fields. This is an area of intense research activity, and this text provides the advanced undergraduate with a superb introduction to this exciting field. Beginning with a simple diagrammatic approach, the book proceeds through recent advances to areas of current research. Topics including topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations combine to form a coherent and highly developed theory with which to explore and explain the accessible and intuitive problems of knots and surfaces. - ;The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces and of group presentations. Beginning with a simple diagrammatic approach to the study of knots, reflecting the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent advances in our understanding to areas of current research. Topics included are straightforward introductions to topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. These topics combine into a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces. Both as an introduction to several areas of prime importance to the development of pure mathematics today, and as an account of pure mathematics in action in an unusual context, this book presents novel challenges to students and other interested readers. -

God and the Universe

Author: Arthur Gibson
Publisher: Routledge
ISBN: 9781136365720
Release Date: 2013-10-11
Genre: Religion

Ambitious, controversial and absorbing, God and the Universe tackles the highly-charged issue of God's relevance in the light of new scientific thinking on cosmology. Engaging with poststructuralism, ethics, mathematics, and philosophy through the ages, this persuasively argued book reinvigorates religious debate for the new millennium.

Representations of Integers as Sums of Squares

Author: E. Grosswald
Publisher: Springer Science & Business Media
ISBN: 9781461385660
Release Date: 2012-12-06
Genre: Mathematics

During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but con sisted of particularly gifted and eager listeners; unfortunately, their back ground varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the inten tion of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be.

Intuitive Topology

Author: Viktor Vasilʹevich Prasolov
Publisher: American Mathematical Soc.
ISBN: 9780821803561
Release Date: 1995
Genre: Mathematics

This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset. Another unusual trait of this book is that it deals mainly with constructions and maps, rather than with proofs that certain maps and constructions do or do not exist. The numerous illustrations are an essential feature. The book is accessible not only to undergraduates but also to high school students and will interest any reader who has some feeling for the visual elegance of geometry and topology.

Hyperbolic Manifolds

Author: Albert Marden
Publisher: Cambridge University Press
ISBN: 9781316432525
Release Date: 2016-02-01
Genre: Mathematics

Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heavily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of exercises and explorations that both challenge the reader to prove assertions made in the text, and suggest further topics to explore that bring additional insight. There is an extensive index and bibliography.

Proceedings

Author: Patrick J. Bresnahan
Publisher: University of South Carolina
ISBN: 0963353209
Release Date: 1992
Genre: Cartography


So Inclined

Author: Nina Marlane Thyrum Joblon
Publisher:
ISBN: UCAL:C3403956
Release Date: 1997
Genre:


Three Dimensional Geometry and Topology Volume 1

Author: William P. Thurston
Publisher: Princeton University Press
ISBN: 9781400865321
Release Date: 2014-10-31
Genre: Mathematics

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 Poincaré 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.