Using Algebraic Geometry (Graduate Texts in Mathematics, 185)

The discovery of new algorithms for dealing with polynomial equations, and their implementation on fast, inexpensive computers, has revolutionized algebraic geometry and led to exciting new applications in the field. This book details many uses of algebraic geometry and highlights recent applications of Grobner bases and resultants. This edition contains two new sections, a new chapter, updated references and many minor improvements throughout.

Computers & Technology

Cox, David A.

O'Shea, Donal

Little,John

Computer Science

Springer

For courses currently engaged, or leaning toward calculus reform. Callahan fully embraces the calculus reform movement in technology and pedagogy, while taking it a step further with a unique organization and applications to real-world problems.

Calculus in Context

Science & Math

CALLAHAN, James

Hoffman, Kenneth R.

Pollatsek, Harriet

Senechal, Lester

Mathematics

W.H.Freeman

The Poincare Conjecture: In Search of the Shape of the Universe

Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting. The Poincaré conjecture lies at the heart of modern geometry and topology, and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point. 
Poincaré's conjecture is one of the seven "millennium problems" that bring a one-million-dollar award for a solution. Grigory Perelman, a Russian mathematician, has offered a proof that is likely to win the Fields Medal, the mathematical equivalent of a Nobel prize, in August 2006. He also will almost certainly share a Clay Institute millennium award. 
In telling the vibrant story of The Poincaré Conjecture, Donal O'Shea makes accessible to general readers for the first time the meaning of the conjecture, and brings alive the field of mathematics and the achievements of generations of mathematicians whose work have led to Perelman's proof of this famous conjecture.

History

Walker Books

"O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman."-Wall Street Journal In 1904, Henri Poincaré, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the Poincaré conjecture, a tantalizing puzzle that speaks to the possible shape of the universe. For more than a century, the conjecture resisted attempts to prove or disprove it. As Donal O'Shea reveals in his elegant narrative, Poincaré's conjecture opens a door to the history of geometry, from the Pythagoreans of ancient Greece to the celebrated geniuses of the nineteenth-century German academy and, ultimately, to a fascinating array of personalities-Poincaré and Bernhard Riemann, William Thurston and Richard Hamilton, and the eccentric genius who appears to have solved it, Grigory Perelman. The solution seems certain to open up new corners of the mathematical universe.

The Poincare Conjecture tells the story behind one of the world's most confounding mathematical theories. Formulated in 1904 by Henri Poincare, his Conjecture promised to describe the very shape of the universe, but remained unproved until a huge prize was offered for its solution in 2000. Six years later, an eccentric Russian mathematician had the answer.Here, Donal O'Shea explains the maths behind the Conjecture and its proof, and illuminates the curious personalities surrounding this perplexing conundrum, along the way taking in a grand sweep of scientific history from the ancient Greeks to Christopher Columbus. This is an enthralling tale of human endeavour, intellectual brilliance and the thrill of discovery.

Allen Lane.

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry―the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz―this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).
The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple™, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.
Readers who are teaching from Ideals, Varieties, and Algorithms, or are studying the book on their own, may obtain a copy of the solutions manual by sending an email to jlittle@holycross.edu.
From the reviews of previous editions:
 “…The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. …The book is well-written. …The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.” 
 ―Peter Schenzel, zbMATH, 2007
 “I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.” 
 ―The American Mathematical Monthly

El matematico frances Henri Poincare (18541912) no solo realizo aportaciones decisivas en el campo de la topologia la ciencia que estudia las propiedades de las formas geometricas, sino que, ademas, lego a la posteridad uno de los problemas matematicos mas fascinantes de todos los tiempos, pues su respuesta puede contribuir a explicar la forma del universo. Desde 1904, lo que se conoce como «conjetura de Poincare» ha desafiado a varias generaciones de investigadores, que han tratado infructuosamente de resolverla o refutarla.Con grandes dotes divulgativas, Donal OShea describe la trayectoria del saber geometrico desde los comienzos en Babilonia y Grecia hasta el presente, cuenta las vicisitudes de personalidades geniales como Euclides, Gauss o Riemann, y, sobre todo, relata el apasionante colofon de este enigma matematico: en efecto, en 2000, el Clay Mathematics Institute declaro la conjetura como uno de los siete problemas fundamentales irresueltos del milenio, y ofrecio un millon de dolares de premio a quien lo solucionase. En 2003, el matematico ruso Grigory Perelmann poco amante de la fama y que en 2006 se permitio rechazar la Medalla Fields, el Nobel de las matematicas asombro a la comunidad cientifica colgando en Internet una serie de articulos que parecian solucionar, finalmente, la conjetura. Un libro sorprendente sobre una autentica odisea intelectual.

La conjetura de Poincaré: En busca de la forma del universo (Metatemas) (Spanish Edition)

Tusquets Editores S.A.

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Books by O'Shea, Donal

See the best price to sell, buy, or rent books by O'Shea, Donal. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Using Algebraic Geometry (Graduate Texts in Mathematics, 185)

Calculus in Context

The Poincare Conjecture: In Search of the Shape of the Universe

The Poincare Conjecture: In Search of the Shape of the Universe

The Poincare Conjecture: In Search of the Shape of the Universe

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

La conjetura de Poincaré: En busca de la forma del universo (Metatemas) (Spanish Edition)