Markov's Theorem and 100 Years of the Uniqueness Conjecture

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, Markov's 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: Markov's 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 Markov's 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)

• 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.​