Book Description

An important new perspective on AFFINE AND PROJECTIVE GEOMETRY

This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view.

Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises, covering material for two semesters of upper-level undergraduate mathematics. The first part of the book deals with the correlation between synthetic geometry and linear algebra. In the second part, geometry is used to introduce lattice theory, and the book culminates with the fundamental theorem of projective geometry.

While emphasizing affine geometry and its basis in Euclidean concepts, the book:

• Builds an appreciation of the geometric nature of linear algebra
• Expands students' understanding of abstract algebra with its nontraditional, geometry-driven approach
• Demonstrates how one branch of mathematics can be used to prove theorems in another
• Provides opportunities for further investigation of mathematics by various means, including historical references at the ends of chapters

Throughout, the text explores geometry's correlation to algebra in ways that are meant to foster inquiry and develop mathematical insights whether or not one has a background in algebra. The insight offered is particularly important for prospective secondary teachers who must major in the subject they teach to fulfill the licensing requirements of many states. Affine and Projective Geometry's broad scope and its communicative tone make it an ideal choice for all students and professionals who would like to further their understanding of things mathematical. ... Read more

Book Description

Physics was transformed between 1890 and 1930, and this volume provides a detailed history of the era and emphasizes the key role of geometrical ideas. Topics include the application of n-dimensional differential geometry to mechanics and theoretical physics, the philosophical questions on the reality of geometry, and the nature of geometry and its connections with psychology, special relativity, Hilbert's efforts to axiomatize relativity, and Emmy Noether's work on physics. ... Read more

Book Description

Hyperbolic Manifolds and Discrete Groups is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout is on the "Big Monster," i.e., on Thurston's hyperbolization theorem, which has not only completely changed the landscape of 3-dimensional topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book presents the first complete proof of Thurston's hyperbolization theorem in the "generic case" and an outline of Otal's proof of the hyperbolization theorem for manifolds fibered over the circle.This important work contains an extended treatment of the theory of Kleinian groups and group actions on trees, including such key topics as:* the Kazhdan--Margulis--Zassenhaus theorem* the Klein and Maskit combination theorems* the Mostow rigidity theorem* the Douady--Earle extension theorem for homeomorphisms of the circle* the smoothness theorem for representation varieties of Kleinian groups* the Ahlfors finiteness theorem* the Brooks deformation theorem* characterization of pseudo-Anosov homeomorphisms* compactification of character varieties via group actions on trees ... Read more

Book Description

This is the first comprehensive treatment of the geometry of complex hyperbolic space, a rich area of research with numerous connections to other branches of mathematics, including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie groups, and harmonic analysis. ... Read more

Amazon.com

What's so sacred about parallel lines? Students and general readers who want a solid grounding in the fundamentals of space would do well to let M. Helena Noronha's Euclidean and Non-Euclidean Geometries be their guide. Noronha, professor of mathematics at California State University, Northridge, breaks geometry down to its essentials and shows students how Riemann, Lobachevsky, and the rest built their own by re-evaluating the parallel postulate. Each chapter devotes itself to rigorous study of one topic: neutral geometry, Euclidean 3-space, hyperbolic geometry, and more reveal themselves to the reader through the author's clear analyses and proofs. Problem sets help the student become comfortable with techniques and reach the conclusions through their own work, gaining a visceral understanding impossible through passive reading. Little mathematical background is needed beyond a bit of set theory, calculus, and a willingness to persevere. --Rob Lightner ... Read more

Book Description

Growing interest in the use of symmetry methods in applied mathematics has triggered this comprehensive overview of the differential geometric view of the subject. Beginning with a background chapter on calculus on manifolds, the book proceeds to more advanced topics and applications, especially concerning singularities of the Monge-Ampere equations that describe phenomena in geophysical fluid dynamics. Many applications areas are covered, and computer code is included for implementing some of the techniques described. ... Read more

Book Description

The study of asymptotic solutions to nonlinear systems of partialdifferential equations is a very powerful tool in the analysisof such systems and their applications in physics, mechanics,and engineering. In the present book, the authors propose a newpowerful method of asymptotic analysis of solutions, which canbe successfully applied in the case of the so-called "smoothedshock waves", i.e., nonlinear waves which vary fast in aneighborhood of the front and slowly outside of thisneighborhood. The proposed method, based on the study ofgeometric objects associated to the front, can be viewed as ageneralization of the geometric optics (or WKB) method forlinear equations. This volume offers to a broad audience asimple and accessible presentation of this new method.

The authors present many examples originating from problems ofhydrodynamics, nonlinear optics, plasma physics, mechanics ofcontinuum, and theory of phase transitions (free boundaryproblems). In the examples, characterized by smoothing ofsingularities due to dispersion or diffusion, asymptoticsolutions in the form of distorted solitons, kinks, breathers,or smoothed shock waves are constructed. By a unified rule, ageometric picture is associated with each physical problem thatallows for obtaining tractable asymptotic formulas and providesa geometric interpretation of the physical process. Included aremany figures illustrating the various physical effects. ... Read more

Book Description

A theory of generalized Cauchy-Riemann systems with polar singularities of order not less than one is presented and its application to study of infinitesimal bending of surfaces having positive curvature and an isolated flat point is given. The book contains results of investigations obtained by the author and his collaborators. ... Read more

Book Description

This unique book gives an informal introduction into the non-Euclidean geometries through a series of dialogues between a somewhat grown-up Alice (of Looking Glass fame), her uncle Lewis Carroll, and a visitor from the twentieth century, Dr Whatif. In the story, Lewis Carroll's geometrical beliefs are cast into the Euclidean mould, Dr Whatif asks the penetrating and controversial questions, and Alice acts as a mediator and interested participant. The book is intentionally more mathematical than Lewis Carroll's books, but for those of us who enjoyed Alice's earlier adventures there are many interesting flashbacks to those inimitable characters: the Red Queen, Tweedle-Dum and his twin brother, the Mad Hatter ... The text is filled with humour, wit, and verses of poetry. Part 1 contains the story in six chapters, each of which concludes with a problem set; Part 2 is more mathematical, and looks at the axiom systems, and gives solutions to the problems. The presentation, with its old-time borders, script headings, and cartoon drawings evokes the spirit of the original Alice. ... Read more

Reviews (1)

Another magical journey through the wonderland of geometry
One cannot say that Lewis Carroll would be pleased, since he helps with the narration of this account of some additional travels of his darling Alice. Many of the other original Wonderland characters also help Dr. WhatIf and him explain and illustrate a wide variety of geometric problems.
Although the author does an excellent job in presenting difficultconcepts, this is not a textbook, Thinkof it as an excursion into thewonderland of advanced geometry. Experts will frolic, amateurs with grinand eventually understand and beginners will smile right up to the point(pun intended), where they get lost.
Problems are given at the end ofeach chapter and detailed solutions appear in the back of the book. Apartial list of the topics covered includes inversions, reflections, ahyperbolic T-party, hyperbolic geometry, and projections. Verse with adecidedly Carrollean flavor appears throughout the text.
A joy toread and even better the second time through, it is a fitting sequel to theoriginal adventures.

Published in Journal of RecreationalMathematics, reprinted with permission. ... Read more

Book Description

Book Description

Book Description

This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry. ... Read more