MAA Press: An Imprint of the American Mathematical Society
MS/MAA Textbooks, Volume: 45;
2018; Hardcover
MSC: Primary 01;
Print ISBN: 978-1-4704-4352-8
The History of Mathematics: A Source-Based Approach is a comprehensive history of the development of mathematics. This, the first volume of the two-volume set, takes readers from the beginning of counting in prehistory to 1600 and the threshold of the discovery of calculus. It is notable for the extensive engagement with original?primary and secondary?source material. The coverage is worldwide, and embraces developments, including education, in Egypt, Mesopotamia, Greece, China, India, the Islamic world and Europe. The emphasis on astronomy and its historical relationship to mathematics is new, and the presentation of every topic is informed by the most recent scholarship in the field.
The two-volume set was designed as a textbook for the authors' acclaimed year-long course at the Open University. It is, in addition to being an innovative and insightful textbook, an invaluable resource for students and scholars of the history of mathematics. The authors, each among the most distinguished mathematical historians in the world, have produced over fifty books and earned scholarly and expository prizes from the major mathematical societies of the English-speaking world.
Undergraduate and graduate students interested in the history of mathematics.
Mathematical Surveys and Monographs, Volume: 40;
2018; 488 pp; Hardcover
MSC: Primary 20;
Print ISBN: 978-1-4704-4189-0
This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series?the completion of the proof of the following theorem:
Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups.
Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.
Graduate students and researchers interested in the theory of finite groups.
MAA Press: An Imprint of the American Mathematical Society
AMS/MAA Textbooks, Volume: 44;
2018; 480 pp; Hardcover
MSC: Primary 51;
Print ISBN: 978-1-4704-4843-1
Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context.
The line and the circle are the principal characters driving the narrative. In every geometry considered?which include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries?these two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area?
There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructibility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.
Undergraduate students interested in geometry.
Graduate Studies in Mathematics, Volume: 197;
2018; 205 pp; Hardcover
MSC: Primary 35; 42;
Print ISBN: 978-1-4704-4292-7
Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrodinger equation, nonlinear hyperbolic conservation laws, and shock waves.
The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.
Undergraduate and graduate students and researchers interested in partial differential equations (PDEs).
Mathematical Surveys and Monographs, Volume: 236;
2018; 427 pp; Hardcover
MSC: Primary 37; 11; 28; 47;
Print ISBN: 978-1-4704-4780-9
Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields.
Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results.
The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.
Graduate students and researchers interested in ergodic theory and its connections to combinatorics and number theory.