Markov's Theorem and 100 Years of the Uniqueness Conjecture
A Mathematical Journey from Irrational Numbers to Perfect Matchings
2013, X, 257 p. 72 illus.
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First comprehensive book on the Theorem of Markov and the Uniqueness Conjecture
Fresh and interdisciplinary approach presenting also the many connections to other fields
Leisurely but rigorous introduction to the "Markov Theme" for students and professional mathematicians
This book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markov’s theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markov’s 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.
Content Level »Research
Keywords »Continued Fractions - Lagrange Spectrum - Markov Theorem - Modular Group - Uniqueness Conjecture of Markov Numbers
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.