Mathematical Surveys and Monographs, Volume: 214
2016; 515 pp; Hardcover
MSC: Primary 35;
Print ISBN: 978-1-4704-2857-0
In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation.
In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he provides a sharp description of the blow-up. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that small-data solutions are global when the null condition is satisfied.
Readers who master the material will have acquired tools on the cutting edge of PDEs, fluid mechanics, hyperbolic conservation laws, wave equations, and geometric analysis.
Mathematical Surveys and Monographs, Volume: 215
2016; 174 pp; Hardcover
MSC: Primary 20;
Print ISBN: 978-1-4704-2901-0
Richard Thompson's famous group F - See more at: http://bookstore.ams.org/surv-215/#sthash.N1HKBokl.dpuf
has the striking property that it can be realized as a dense subgroup of the group of all orientation-preserving homeomorphisms of the unit interval, but it can also be given by a simple 2-generator-2-relator presentation, in fact as the fundamental group of an aspherical complex with only two cells in each dimension.
This monograph studies a natural generalization of F
that also includes Melanie Stein's generalized F -groups. The main aims of this monograph are the determination of isomorphisms among the generalized F
groups and the study of their automorphism groups.
This book is aimed at graduate students (or teachers of graduate students) interested in a class of examples of torsion-free infinite groups with elements and composition that are easy to describe and work with, but have unusual properties and surprisingly small presentations in terms of generators and defining relations.
Contemporary Mathematics
Volume: 677; 2016; 229 pp; Softcover
MSC: Primary 20; 68;
Print ISBN: 978-1-4704-2303-2
This volume contains the proceedings of three special sessions: Algebra and Computer Science, held during the Joint AMS-EMS-SPM meeting in Porto, Portugal, June 10-13, 2015; Groups, Algorithms, and Cryptography, held during the Joint Mathematics Meeting in San Antonio, TX, January 10?13, 2015; and Applications of Algebra to Cryptography, held during the Joint AMS-Israel Mathematical Union meeting in Tel-Aviv, Israel, June 16?19, 2014.
Papers contained in this volume address a wide range of topics, from theoretical aspects of algebra, namely group theory, universal algebra and related areas, to applications in several different areas of computer science.
From the computational side, the book aims to reflect the rapidly emerging area of algorithmic problems in algebra, their computational complexity and applications, including information security, constraint satisfaction problems, and decision theory.
The book gives special attention to recent advances in quantum computing that highlight the need for a variety of new intractability assumptions and have resulted in a new area called group-based cryptography.
Contemporary Mathematics, Volume: 678
2016; 316 pp; Softcover
MSC: Primary 37; 01;
Print ISBN: 978-1-4704-2299-8
This volume contains the proceedings of three conferences in Ergodic Theory and Symbolic Dynamics: the Oxtoby Centennial Conference, held from October 30?31, 2010, at Bryn Mawr College; the Williams Ergodic Theory Conference, held from July 27?29, 2012, at Williams College; and the AMS Special Session on Ergodic Theory and Symbolic Dynamics, held from January 17-18, 2014, in Baltimore, MD.
This volume contains articles covering a variety of topics in measurable, symbolic and complex dynamics. It also includes a survey article on the life and work of John Oxtoby, providing a source of information about the many ways Oxtoby's work influenced mathematical thought in this and other fields.
Student Mathematical Library, Volume: 80
2016; 343 pp; Softcover
MSC: Primary 91;
Print ISBN: 978-1-4704-2210-3
This book offers a gentle introduction to the mathematics of both sides of game theory: combinatorial and classical. The combination allows for a dynamic and rich tour of the subject united by a common theme of strategic reasoning.
The first four chapters develop combinatorial game theory, beginning with an introduction to game trees and mathematical induction, then investigating the games of Nim and Hackenbush. The analysis of these games concludes with the cornerstones of the Sprague-Grundy Theorem and the Simplicity Principle.
The last eight chapters of the book offer a scenic journey through the mathematical highlights of classical game theory. This contains a thorough treatment of zero-sum games and the von Neumann Minimax Theorem, as well as a student-friendly development and proof of the Nash Equilibrium Theorem. The Folk Theorem, Arrow's voting paradox, evolutionary biology, cake cutting, and other engaging auxiliary topics also appear.
The book is designed as a textbook for an undergraduate mathematics class. With ample material and limited dependencies between the chapters, the book is adaptable to a variety of situations and a range of audiences. Instructors, students, and independent readers alike will appreciate the flexibility in content choices as well as the generous sets of exercises at various levels.
Undergraduate students, graduate students, and researchers interested in game theory.
Graduate Studies in Mathematics, Volume: 175
2016; 455 pp; Hardcover
MSC: Primary 35; 37; 53; 58;
Print ISBN: 978-1-4704-0986-9
Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs.
Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G
structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence.
This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.
Graduate students and researchers interested in differential geometry, in particular, in exterior systems and the moving frames method and in its applications in algebraic geometry, PDE, and other areas of mathematics