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Afterword by Herbert A. Hauptman, Nobel Laureate

The Pythagorean theorem may be the best-known equation in mathematics. Its origins reach back to the beginnings of civilization, and today every student continues to study it. What most nonmathematicians don't understand or appreciate is why this simply stated theorem has fascinated countless generations.

In this entertaining and informative book, veteran math educator Alfred S. Posamentier makes the importance of the Pythagorean theorem delightfully clear.

He begins with a brief history of Pythagoras and the early use of his theorem by the ancient Egyptians, Babylonians, Indians, and Chinese, who used it intuitively long before Pythagoras's name was attached to it.

He then shows the many ingenious ways in which the theorem has been proved visually using highly imaginative diagrams. Some of these go back to ancient mathematicians; others are comparatively recent proofs, including one by the twentieth president of the United States, James A. Garfield.

After demonstrating some curious applications of the theorem, Posamentier then explores the Pythagorean triples, pointing out the many hidden surprises of the three numbers that can represent the sides of the right triangle (e.g, 3, 4, 5 and 5, 12, 13). And many will truly amaze the reader.

He then turns to the "Pythagorean means" (the arithmetic, geometric, and harmonic means). By comparing their magnitudes in a variety of ways, he gives the reader a true appreciation for these mathematical concepts.

The final two chapters view the Pythagorean theorem from an artistic point of view--namely, how Pythagoras's work manifests itself in music and how the Pythagorean theorem can influence fractals.

Posamentier's lucid presentation and gift for conveying the significance of this key equation to those with little math background will inform, entertain, and inspire the reader, once again demonstrating the power and beauty of mathematics! ... Read more

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A fine addition to any math history library
The Pythagorean Theorem: The Story of its Power and Beauty offers high school to college-level collections a fine survey of the best-known equation in math, which every math student studies. Why is it fascinating? Veteran math educator Alfred S. Posamentier offers a fine history of the theorem and its importance over the centuries, linking it to mankind's development and evolution and exploring its hidden aspects. A fine addition to any math history library.
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This two volume work on ... Read more

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This two volume work on ... Read more

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This text examines topological methods in algebraic geometry. ... Read more

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The most significant recent development in number theory is the work of Andrew Wiles on modular elliptic curves. Besides implying Fermat's Last Theorem, his work establishes a new reciprocity law. Reciprocity laws lie at the heart of number theory.

Wiles' work draws on many of the tools of modern number theory and the purpose of this volume is to introduce readers to some of this background material.

Based on a seminar held during 1993--1994 at the Fields Institute for Research in Mathematical Sciences, this book contains articles on elliptic curves, modular forms and modular curves, Serre's conjectures, Ribet's theorem, deformations of Galois representations, Euler systems, and annihilators of Selmer groups. All of the authors are well known in their field and have made significant contributions to the general area of elliptic curves, Galois representations, and modular forms.

Features:

Brings together a unique collection of number theoretic tools.

Makes accessible the tools needed to understand one of the biggest breakthroughs in mathematics.

Provides numerous references for further study. ... Read more

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Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies. ... Read more

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Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. The author's original research in queuing theory and dynamical systems figures prominently in the book. This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards. The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems. The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts. ... Read more

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Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras---such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory---on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics.

This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications.

Features:

Contributions by originators of the subject who are leaders in the field.

Survey articles not previously available.

Expository articles geared toward the larger mathematical community. ... Read more

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This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Customer Reviews (1)

Great book.
I am no expert in this field, but I do not understand why this great book has not gotten wider exposure.The author gives a wonderful intro to the "geometry of compact Riemann surfaces based on hyperbolic geometry and on cutting and pasting."(Quoting from the preface.)

Perhaps Amazon has not helped his cause by miss-listing the author as J. Buser instead of Peter Buser, and compounding the problem by not returning this book on a search by author for Peter Buser.

All in all, this (fairly) advanced amateur gives Peter Buser's book on Riemann Surfaces five stars.
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The first half of this work gives a treatment of Deligne's functorial intersection theory tailored to the needs of this paper. This treatment is intended to satisfy three requirements: 1) that it be general enough to handle families of singular curves, 2) that it be reasonably self-contained, and 3) that the constructions given be readily adaptable to the process of adding norms and metrics such as is done in the second half of the paper.

The second half of the work is devoted to developing a class of intersection functions for singular curves that behaves analogously to the canonical Green's functions introduced by Arakelov for smooth curves. These functions are called intersection functions since they give a measure of intersection over the infinite places of a number field. The intersection over finite places can be defined in terms of the standard apparatus of algebraic geometry.

Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem. ... Read more

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This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics include: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills functional; geodesics and Jacobi fields, Rauch comparison theorem and applications; Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics; symmetric spaces and Kähler manifolds; the Palais-Smale condition and closed geodesics; Harmonic maps, minimal surfaces. ... Read more

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maths background for General Relativity and QFT
For theoretical physicists, especially those studying Einstein's Theory of General Relativity, Or if your subject is quantum field theory. Jost's book is good preparation. He offers an in-depth teaching of Riemannian geometry. So ideas like covariant and contravariant derivatives on a manifold take on elegant meaning.

Note that General Relativity does not get an explicit mention. However, a typical physics GR course might often not have time to give a good discussion of the underlying maths. And standard GR texts, like Misner, Thorne and Wheeler or Weinberg, also tend to have very abbreviated explanations of the maths. So Jost's book is useful for those of you inclined to look further.

The length of the book means it's probably too long for a standard 1 term or semester course, if the intent is to entirely cover the book.

Intro to Riemannian Geom. and Geom. Analysis
Covers standard material on Reimannian Geometry. In addition: variational problems from QFT. Spin geometry and Dirac operators are explained in detail. ... Read more

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Chapters: Convex Set, Minkowski's Theorem, Face, Convex Hull, Antimatroid, Convex Polytope, Radon's Theorem, Oriented Matroid, Convex Hull Algorithms, Minkowski Addition, Graham Scan, Helly's Theorem, Orthogonal Convex Hull, Rotating Calipers, Mahler Volume, Convex Metric Space, Carathéodory's Theorem, Separating Axis Theorem, Chan's Algorithm, Tangent Cone, Gift Wrapping Algorithm, Brunn-minkowski Theorem, Dual Cone and Polar Cone, Kirkpatrick-seidel Algorithm, Conical Combination, Krein-milman Theorem, Convex Combination, Convex and Concave Polygons, Supporting Hyperplane, Geodesic Convexity, Gilbert-johnson-keerthi Distance Algorithm, M. Riesz Extension Theorem, Tverberg's Theorem, B-Convex Space, Minkowski Portal Refinement, Absolutely Convex Set, List of Convexity Topics, Kakutani's Theorem, Concave Set. Source: Wikipedia. Pages: 144. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts; see Korte et al. (1991) for a comprehensive survey of antimatroid theory with many additional references. The axioms defining antimatroids as set systems have some similarity to those of matroids, but whereas matroids can be defined by an exchange axiom (e.g., the basis exchange, or inde...More: http://booksllc.net/?id=667063 ... Read more

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Chapters: Analytic Geometry, Homological Algebra, Bézout's Theorem, Generalized Riemann Hypothesis, Motive, Dessin D'enfant, Glossary of Arithmetic and Diophantine Geometry, Hodge Structure, Smooth Function, Nakayama Lemma, Resolution of Singularities, Algebraic K-Theory, Grassmannian, Valuation, Gröbner Basis, Calabi-yau Manifold, Chow Ring, Cotangent Complex, Algebraic Variety, Crystalline Cohomology, Esquisse D'un Programme, Macdonald Polynomial, Hilbert Scheme, Flat Module, Algebraic Geometry and Analytic Geometry, Homogeneous Polynomial, Quantum Cohomology, Gromov-witten Invariant, Drinfel'd Module, General Position, Ample Line Bundle, Milnor Number, Flip, Generic Property, Minimal Model Program, Hilbert's Arithmetic of Ends, Regular Local Ring, Adjunction Formula, Néron-tate Height, Pseudoholomorphic Curve, Codimension, Injective Sheaf, Characteristic Set, Kähler Manifold, Conifold, Dévissage, Rational Mapping, Arakelov Theory, Hasse-weil Zeta Function, Cohen-macaulay Ring, Algebraic Stack, List of Algebraic Geometry Topics, Lefschetz Theorem on (1,1)-Classes, A¹ Homotopy Theory, Adequate Equivalence Relation, Italian School of Algebraic Geometry, Fondements de La Géometrie Algébrique, Standard Conjectures on Algebraic Cycles, Shimura Variety, Algebraic Space, Cone of Curves, Hyperkähler Manifold, Deformation Theory, Noncommutative Algebraic Geometry, Flat Topology, Descent, Excellent Ring, Stable Vector Bundle, Finite Morphism, Amoeba, Tropical Geometry, Canonical Singularity, Complete Intersection, Tangent Cone, N!-Conjecture, Conic Bundle, Irreducible Component, Canonical Ring, Residue Field, Abstract Algebraic Variety, Inverse Image Functor, Kähler Differential, Formal Scheme, Grothendieck's Galois Theory, Mnev's Universality Theorem, Jacobian Conjecture, Local Parameter, Period, Differential of the First Kind, Positive Form, Zariski Tangent Space, Schubert Variety, Generic Point, Noetherian Topological Space, Néron ...More: http://booksllc.net/?id=1997 ... Read more

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Publisher: Boston, D.C. Heath ... Read more