Markov's Theorem and 100 Years of the Uniqueness Conjecture

Markov's Theorem and 100 Years of the Uniqueness Conjecture
Author: Martin Aigner
Publisher: Springer Science & Business Media
Total Pages: 257
Release: 2013-07-18
Genre: Mathematics
ISBN: 3319008889

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.

An Illustrated Theory of Numbers

An Illustrated Theory of Numbers
Author: Martin H. Weissman
Publisher: American Mathematical Soc.
Total Pages: 341
Release: 2020-09-15
Genre: Education
ISBN: 1470463717

News about this title: — Author Marty Weissman has been awarded a Guggenheim Fellowship for 2020. (Learn more here.) — Selected as a 2018 CHOICE Outstanding Academic Title — 2018 PROSE Awards Honorable Mention An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.

Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry

Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry
Author: Sergey Novikov
Publisher: American Mathematical Soc.
Total Pages: 480
Release: 2021-04-12
Genre: Education
ISBN: 1470455927

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

From Christoffel Words to Markoff Numbers

From Christoffel Words to Markoff Numbers
Author: Christophe Reutenauer
Publisher:
Total Pages: 169
Release: 2019
Genre: Mathematics
ISBN: 0198827547

This book looks to expand on the relationship between Christoffel words and Markoff theory. Part 1 focuses on the classical theory of Markoff, while part II explores the more advanced and recent results around Christoffel words.

Armenian Organization and Ideology under Ottoman Rule

Armenian Organization and Ideology under Ottoman Rule
Author: Dikran Mesrob Kaligian
Publisher: Transaction Publishers
Total Pages: 273
Release: 2011-12-31
Genre: Political Science
ISBN: 1412848342

This book provides a comprehensive picture of Armeno-Turkish relations for the brief period of Ottoman Constitutional rule between 1908 and 1914. Kaligian integrates internal documents of the Armenian Revolutionary Federation, and existing research on the last years of the empire, as well as the archives of the British, American, and German diplomatic corps. By reducing the overemphasis on central government policies and by describing unofficial contacts, political relations, and provincial administration and conditions, Kaligian provides a unified account of this key period in Ottoman history. Kaligian sets out to resolve many of the conflicting conclusions in the current historiography—including the most central issue, the Armenian Revolutionary Federation relations with the Turkish Committee of Union and Progress. It is impossible to obtain a true picture of Armeno-Turkish relations without an accurate analysis of their two leading parties. This study finds that the ARF was torn between maintaining relations with a CUP that had failed to implement promised reforms and was doing little to prevent increasing attacks on the Armenian population, or break off relations thus ending any realistic chance for the constitutional system to succeed. The party continued to stake its reputation and resources on the success of constitutional government even after the trauma of the 1909 Adana massacres. The decisive issue was the failure of land restitution. This book sets the record straight in terms of understanding Armeno-Turkish relations during this short but pivotal period. Kaligian’s study, the first of its kind, shows that the party’s internal deliberations support the conclusion that it did remain loyal and contradicts the view that the party’s only aim was to incite a rebellion against Ottoman rule. The author has done an excellent job of leading the reader through this rich history, using primary source information to bridge the gaps from theory, to analysis, to evidence.

Combinatorics on Words

Combinatorics on Words
Author: Robert Mercaş
Publisher: Springer Nature
Total Pages: 340
Release: 2019-09-02
Genre: Mathematics
ISBN: 3030287963

This book constitutes the refereed proceedings of the 12th International Conference on Combinatorics on Words, WORDS 2019, held in Loughborough, UK, in September 2019. The 21 revised full papers presented in this book together with 5 invited talks were carefully reviewed and selected from 34 submissions. WORDS is the main conference series devoted to the mathematical theory of words. In particular, the combinatorial, algebraic and algorithmic aspects of words are emphasized. Motivations may also come from other domains such as theoretical computer science, bioinformatics, digital geometry, symbolic dynamics, numeration systems, text processing, number theory, etc.

Farey Sequences

Farey Sequences
Author: Andrey O. Matveev
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 182
Release: 2017-11-07
Genre: Mathematics
ISBN: 311054766X

As a first comprehensive overview on Farey sequences and subsequences, this monograph is intended as a reference for anyone looking for specific material or formulas related to the subject. Duality of subsequences and maps between them are discussed and explicit proofs are shown in detail. From the Content Basic structural and enumerative properties of Farey sequences, Collective decision making, Committee methods in pattern recognition, Farey duality, Farey sequence, Fundamental Farey subsequences, Monotone bijections between Farey subsequences

The Knot Book

The Knot Book
Author: Colin Conrad Adams
Publisher: American Mathematical Soc.
Total Pages: 330
Release: 2004
Genre: Mathematics
ISBN: 0821836781

Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.