Student Mathematical Library, Volume: 88
2019; 239 pp; Softcover
MSC: Primary 11; 12;
Print ISBN: 978-1-4704-4399-3
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists.
This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Godel's incompleteness theorems.
Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
Undergraduate and graduate students and researchers interested in number theory and logic.
Contemporary Mathematics, Volume: 727
2019; 355 pp; Softcover
MSC: Primary 16; 11; 13; 08;
Print ISBN: 978-1-4704-4104-3
This book contains the proceedings of the Fifth International Conference on Noncommutative Rings and their Applications, held from June 12?15, 2017, at the University of Artois, Lens, France.
The papers are related to noncommutative rings, covering topics such as: ring theory, with both the elementwise and more structural approaches developed; module theory with popular topics such as automorphism invariance, almost injectivity, ADS, and extending modules; and coding theory, both the theoretical aspects such as the extension theorem and the more applied ones such as Construction A or Reed?Muller codes. Classical topics like enveloping skewfields, weak Hopf algebras, and tropical algebras are also presented.
Graduate students and research mathematicians interested in ring theory and coding theory.
Contemporary Mathematics,Volume: 728
2019; 194 pp; Softcover
MSC: Primary 16; 17; 18; 20;
Print ISBN: 978-1-4704-4321-4
This volume contains the proceedings of the scientific session “Hopf Algebras and Tensor Categories”, held from July 27?28, 2017, at the Mathematical Congress of the Americas in Montreal, Canada.
Papers highlight the latest advances and research directions in the theory of tensor categories and Hopf algebras. Primary topics include classification and structure theory of tensor categories and Hopf algebras, Gelfand-Kirillov dimension theory for Nichols algebras, module categories and weak Hopf algebras, Hopf Galois extensions, graded simple algebras, and bialgebra coverings.
Graduate students and research mathematicians interested in Hopf algebras, tensor categories, and related areas.
Pure and Applied Undergraduate Texts, Volume: 36
2019; Hardcover
MSC: Primary 26;
Print ISBN: 978-1-4704-4928-5
This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions.
A theme in the book is to give more than one proof for interesting facts; this illustrates how different ideas interact and it makes connections among the facts that are being learned. Metric spaces are introduced early in the book, but there are instructions on how to avoid metric spaces for the instructor who wishes to do so. There are questions that check the readers' understanding of the material, with solutions provided at the end. Topics that could be optional or assigned for independent reading include the Cantor function, nowhere differentiable functions, the Gamma function, and the Weierstrass theorem on approximation by continuous functions.
Undergraduate and graduate students interested in learning and teaching undergraduate real analysis.
Proceedings of Symposia in Pure Mathematics, Volume: 102
2019; 282 pp; Hardcover
MSC: Primary 57; 20; 53; 55;
Print ISBN: 978-1-4704-4249-1
This volume contains the proceedings of the 2017 Georgia International Topology Conference, held from May 22?June 2, 2017, at the University of Georgia, Athens, Georgia.
The papers contained in this volume cover topics ranging from symplectic topology to classical knot theory to topology of 3- and 4-dimensional manifolds to geometric group theory. Several papers focus on open problems, while other papers present new and insightful proofs of classical results.
Taken as a whole, this volume captures the spirit of the conference, both in terms of public lectures and informal conversations, and presents a sampling of some of the great new ideas generated in topology over the preceding eight years.
Graduate students and researchers interested in topology.
CBMS Regional Conference Series in Mathematics, Volume: 131
2019; Softcover
MSC: Primary 33; 39; 14; 32;
Print ISBN: 978-1-4704-5038-0
A co-publication of the AMS and CBMS
Discrete Painleve equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics.
This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
Graduate students and researchers interested in integrable systems, mathematical physics, applied mathematics and special functions, as well as resolution of singularities, dynamical systems, and birational geometry.