Glimpses of Algebra and Geometry (Undergraduate Texts in Mathematics) by Gabor Toth (PDF)

Description

Previous edition sold 2000 copies in 3 years; Explores the subtle connections between Number Theory, Classical Geometry and Modern Algebra; Over 180 illustrations, as well as text and Maple files, are available via the web facilitate understanding: http://mathsgi01.rutgers.edu/cgi-bin/wrap/gtoth/; Contains an insert with 4-color illustrations; Includes numerous examples and worked-out problems

User’s Reviews

Editorial Reviews: Review From the reviews of the second edition:”Toth’s ‘Glimpses’ offer selected material that connect algebra and geometry … . This second edition is a revised and substantially expanded version, so for example it includes a detailed treatment of the solution of the cubic and quartic, as well as a long new chapter on Klein’s famous work on the quintic and the icosahedron.” (Günter M. Ziegler, Zentralblatt MATH, Vol. 1027, 2004)”The book is intended – and really manages it – to fill undergraduates with enthusiasm to reach the graduate level. … the author presents various topics of number theory, geometry and algebra and at the same time shows their connection resp. interplay, thus making the study lively and fascinating for the reader. … information on advanced websites and films show how carefully the author has done his job. So this second edition hopefully will not be the last one.” (G. Kowol, Monatshefte für Mathematik, Vol. 141 (2), 2004)”The text covers a wide range of topics and gives a taste of advanced material in number theory, geometry and algebra, particularly where these fields overlap. … there are plenty of references for the interested reader who wishes to pursue a particular topic in greater depth. … the accessibility of the format and the flow of the material combine to create an entertaining and informative work. I recommend the text as a good read for mathematicians of all specialities. ” (Stephen Lucas, The Australian Mathematical Society Gazette, Vol. 30 (4), 2003)”This is the second, much revised and augmented edition of the book originally published in 1998. It intends to close the gap between undergraduate and graduate studies in number theory, classical geometry and modern algebra. … Each of the chapters is a good read and the book adds up to a wholly appealing entity. … It can be warmly recommended … . I can well imagine that teachers … as well as scientists … will benefit from this carefully worked-out textbook.” (J. Lang, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003) From the Back Cover 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 Kleins “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 Toths main fields of interest involve the geometry of eigenmaps and spherical minimal immersions and the visualization of mathematics via computers.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is a delight from start to finish. The material on möbius transformations and geometry, in particular, is well done and easy to digest. The original idea of indicating the depth of material by card suits is a bit befuddling (what does spade mean again) but my impression is that most of the material will be easily understood by any math major with some analysis and algebra under their belt. The analytic approach to non-Euclidean geometry makes it fairly easy to follow and the connections outlined between elements of classical mathematics are enlightening (and unlikely to be encountered elsewhere). If you are looking for something elementary to read about Riemann surfaces, this is an excellent place to start – especially if you have not encountered differential forms before. A pleasure to dip into again and again!

⭐The books does a very good job bridging the gap between undergrad textbooks and grad textbooks. The only thing that I would have like to seen is a few more examples and problems for the book. Overall it is a very good book and well worth going over all the examples and problems that are in the end of the chapters. An give a good points of view of non-Euclidean geometry, and explains mobius geometry well.

⭐Interesting introduction to geometry with very workable examples; I have found it an enjoyable and casual reading. Moreover, this is not merely a book on geometry but as the title says about geometry and algebra; it seems that the author has tried to avoid well trodden paths in order to present the main topics of the first chapters; especially D’alembert-Gauss theorem and solutions of quartic and cubic equations with radicals; one can also find a bit of number theory: Liouville numbers and the fact that cos(pi/n) is algebraic; further on, Toth takes us up to more elaborate subjects such as Fuchsian groups, a glimpse of Riemann surfaces and finally the platonic solids and Klein’s study of the icosahedron and the quintic equation; he even takes time to introduce Euler’s extraordinary formula: V + F = E + 2 so the book is replete with beautiful subjects going back 150 years ago

⭐I have a copy of the first edition which has an interesting treatment of projective geometry. The 2cd edition appears to cover even more.

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