Markov's Theorem and 100 Years of the Uniqueness Conjecture (häftad)
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Format
Häftad (Paperback / softback)
Språk
Engelska
Antal sidor
257
Utgivningsdatum
2015-08-08
Upplaga
Softcover reprint of the original 1st ed. 2013
Förlag
Springer International Publishing AG
Illustrationer
X, 257 p.
Dimensioner
235 x 155 x 10 mm
Vikt
384 g
Antal komponenter
1
Komponenter
1 Paperback / softback
ISBN
9783319033099

Markov's Theorem and 100 Years of the Uniqueness Conjecture

A Mathematical Journey from Irrational Numbers to Perfect Matchings

av Martin Aigner
Häftad,  Engelska, 2015-08-08
1587
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This book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markovs theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markovs theorem. What makes the Markov theme so attractive is that it appears in an astounding variety of different fields, from number theory to combinatorics, from classical groups and geometry to the world of graphs and words. On the way, there are also introductory forays into some fascinating topics that do not belong to the standard curriculum, such as Farey fractions, modular and free groups, hyperbolic planes, and algebraic words. The book closes with a discussion of the current state of knowledge about the uniqueness conjecture, which remains an open challenge to this day. All the material should be accessible to upper-level undergraduates with some background in number theory, and anything beyond this level is fully explained in the text. This is not a monograph in the usual sense concentrating on a specific topic. Instead, it narrates in five parts Numbers, Trees, Groups, Words, Finale the story of a discovery in one field and its many manifestations in others, as a tribute to a great mathematical achievement and as an intellectual pleasure, contemplating the marvellous unity of all mathematics.
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From the book reviews: The topic, and its presentation, does make it a fine source for seminar usage. this is a fine text for students who are ready to see material connecting various areas of mathematics. It reveals the beauty and hints at the excitement of live mathematics. (Thomas A. Schmidt, Mathematical Reviews, September, 2014) In number theory, Markovs theorem (1879) reveals surprising structure within a set of real numbers, called the Lagrange spectrum, which collects precise information about approximability of irrational numbers. Summing Up: Highly recommended. Upper-division undergraduates and above. (D. V. Feldman, Choice, Vol. 51 (8), April, 2014) Book tells the story of a celebrated theorem and an intriguing conjecture: Markovs theorem from 1879 and the uniqueness conjecture formulated by Frobenius . author takes the opportunity to look at this theorem and this conjecture from many different viewpoints . He offers a journey through the mathematical world around Markovs theorem in a leisurely and relaxed style, making his book very pleasant to read. An undergraduate student will certainly enjoy this reading and learn a lot. (Yann Bugeaud, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 116, 2014) This beautiful book gives readers a chance to familiarize themselves with a very simple and yet very difficult problem in number theory, and teaches them that it pays to look at a problem from many different angles. I recommend it to all students who are already hooked to number theory, and perhaps even more to those who are not. (Franz Lemmermeyer, zbMATH, Vol. 1276, 2014)

Innehållsförteckning

Approximation of Irrational Numbers.- Markov's Theorem and the Uniqueness Conjecture.- The Markov Tree.- The Cohn Tree.- The Modular Group SL(2,Z).- The Free Group F2.- Christoffel Words.- Sturmian Words.- Proof of Markov's Theorem.- The Uniqueness Conjecture.

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