Glimpses of algebra and geometry

Gabor Toth

450 pages, parution le 03/06/2002 (2^eme édition)

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Résumé

The purpose of Glimpses of Algebra and Geometry is to fill a gap between undergraduate and graduate mathematics studies. It is one of the few undergraduate texts to explore the subtle and sometimes puzzling connections between Number Theory, Classical Geometry and Modern Algebra in a clear and easily understandable style. Over 160 computer-generated images, accessible to readers via the World Wide Web, facilitate an understanding of mathematical concepts and proofs even further. Glimpses also sheds light on some of the links between the first recorded intellectual attempts to solve ancient problems of Number Theory and Geometry and twentieth century mathematics. GLIMPSES will appeal to students who wish to learn modern mathematics, but have few prerequisite courses, and to high-school teachers who always had a keen interest in mathematics, but seldom the time to pursue background technicalities. Even postgraduate mathematicians will enjoy being able to browse through a number of mathematical disciplines in one sitting.

This new edition includes invaluable improvements throughout the text, including an in-depth treatment of root formulas, a detailed and complete classification of finite Möbius groups a la Klein, and a quick, direct, and modern approach to Felix Klein "Normalformsatz," the main result of his spectacular theory of icosahedron and his solution of the irreducible quintic in terms of hypergeometric functions. Gabor Toth is the Chair and Graduate Director of the Department of Mathematical Sciences at Rutgers University, Camden. His previous publications include Finite Mobius Groups, Spherical Minimal Immersions and Moduli (2001), Harmonic Maps and Minimal Immersion Through Representation Theory (1990) and Harmonic and Minimal Maps with Applications in Geometry and Physics (1984). Professor Toth main fields of interest involve the geometry of eigenmaps and spherical minimal immersions and the visualization of mathematics via computers.

Contents:

A Number is a multitude composed of units" (Euclid).- "..there are no irrational numbers at all" (Kronecker)
Rationality, Elliptic Curves and Fermat's Last Theorem.- Algebraic or Transcendential ?
Complex Arithmetic
Quadratic, Cubic, and Quartic Equations
Stereographic Projection
Proof of the Fundamental Theorem of Algebra
Symmetries of Regular Polygons
Discrete Subgroups of Iso(R2)
Möbius Geometry
Complex Linear Fractional Transformations.- "Out of nothing I have created a new universe" (Bolyai)
Fuchsian Groups
Riemann Surfaces
General Surfaces
The Five Platonic Solids
Finite Möbius Groups
Detour in Topology: Euler-Poincare Characteristic
Detour in Graph Theory: Euler, Hamilton and the Four Color Theorem
Dimension Leap
Quaternions
Back to R3!
Invariants
The Icosahedron and the Unsolvable Quintic
The Fourth Dimension
Appendices
Solutions for 100 Selected Problems.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Gabor Toth
Parution	03/06/2002
Édition	2^eme édition
Nb. de pages	450
Format	18,2 x 24
Couverture	Broché
Poids	992g
Intérieur	Noir et Blanc
EAN13	9780387953458
ISBN13	978-0-387-95345-8