Miscellaneous Books
2022; 298 pp; Hardcover
MSC: Primary 05; 52;
Print ISBN: 978-1-4704-6897-2
Tiling theory provides a wonderful opportunity to illustrate both the beauty and utility of mathematics. It has all the relevant ingredients: there are stunning pictures; open problems can be stated without having to spend months providing the necessary background; and there is both deep mathematics and applications.
Furthermore, tiling theory happens to be an area where many of the sub-fields of mathematics overlap. Tools can be applied from linear algebra, algebra, analysis, geometry, topology, and combinatorics. As such, it makes for an ideal capstone course for undergraduates or an introductory course for graduate students. This material can also be used for a lower-level course by skipping the more technical sections. In addition, readers from a variety of disciplines can read the book on their own to find out more about this intriguing subject.
This book covers the necessary background on tilings and then delves into a variety of fascinating topics in the field, including symmetry groups, random tilings, aperiodic tilings, and quasicrystals. Although primarily focused on tilings of the Euclidean plane, the book also covers tilings of the sphere, hyperbolic plane, and Euclidean 3-space, including knotted tilings. Throughout, the book includes open problems and possible projects for students. Readers will come away with the background necessary to pursue further work in the subject.
Undergraduate and graduate students and researchers interested in tilings and tessellations.
Graduate Studies in Mathematics Volume: 223;
2022; 695 pp; Hardcover
MSC: Primary 20; 37;
Print ISBN: 978-1-4704-6380-9
Product Code: GSM/223
This book is devoted to group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems. One of the main applications of this approach to group theory is the study of asymptotic properties of groups such as growth and amenability. The book presents recently developed techniques of studying groups of dynamical origin using the structure of their orbits and associated groupoids of germs, applications of the iterated monodromy groups to hyperbolic dynamical systems, topological full groups and their properties, amenable groups, groups of intermediate growth, and other topics. The book is suitable for graduate students and researchers interested in group theory, transformations defined by automata, topological and holomorphic dynamics, and theory of topological groupoids. Each chapter is supplemented by exercises of various levels of complexity.
Graduate students and researchers interested in dynamics on groups and asymptotic properties of groups
Mathematical Surveys and Monographs Volume: 270;
2022; 287 pp; Softcover
MSC: Primary 05; 15;
Print ISBN: 978-1-4704-6655-8
Product Code: SURV/270
This book provides an introduction to the inverse eigenvalue problem for graphs (IEP-G) and the related area of zero forcing, propagation, and throttling. The IEP-G grew from the intersection of linear algebra and combinatorics and has given rise to both a rich set of deep problems in that area as well as a breadth of gancillaryh problems in related areas.
The IEP-G asks a fundamental mathematical question expressed in terms of linear algebra and graph theory, but the significance of such questions goes beyond these two areas, as particular instances of the IEP-G also appear as major research problems in other fields of mathematics, sciences and engineering. One approach to the IEP-G is through rank minimization, a relevant problem in itself and with a large number of applications. During the past 10 years, important developments on the rank minimization problem, particularly in relation to zero forcing, have led to significant advances in the IEP-G.
The monograph serves as an entry point and valuable resource that will stimulate future developments in this active and mathematically diverse research area.
Graduate students and researchers interested in inverse eigenvalue problems for graph and rank minimization.
Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from August 2022
FORMAT: Paperback ISBN: 9781009018586
Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti?Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel?Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.
The first comprehensive introduction to the theory of algebraic group schemes over fields
This book is accessible to non-specialists, with few prerequisites
The book is written in the language of modern algebraic geometry
'All together, this excellent text fills a long-standing gap in the textbook literature on algebraic groups. It presents the modern theory of group schemes in a very comprehensive, systematic, detailed and lucid manner, with numerous illustrating examples and exercises. It is fair to say that this reader-friendly textbook on algebraic groups is the long-desired modern successor to the old, venerable standard primers c' Werner Kleinert, zbMath
'The author invests quite a lot to make difficult things understandable, and as a result, it is a real pleasure to read the book. All in all, with no doubt, Milne's new book will remain for decades an indispensable source for everybody interested in algebraic groups.' Boris E. Kunyavski?, MathSciNet
available from November 2022
FORMAT: HardbackISBN: 9781107191136
This detailed yet accessible text provides an essential introduction to the advanced mathematical methods at the core of theoretical physics. The book steadily develops the key concepts required for an understanding of symmetry principles and topological structures, such as group theory, differentiable manifolds, Riemannian geometry, and Lie algebras. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering. The material has been subject to more than twenty years of feedback from students, ensuring that explanations and examples are lucid and considered, and numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding. This text unites a wide variety of important topics that are often scattered across different books, and provides a solid platform for more specialized study or research.
Unites a wide variety of important topics that are often scattered across different books
Written in a clear, pedagogical style and accessible to students from physics, mathematics, and engineering
Numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding
Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from October 2022
FORMAT: Hardback ISBN: 9781316514887
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered include diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Numerous examples highlight both surprising connections between finite- and infinite-dimensional geometry, and challenges occurring solely in infinite dimensions. The geometric techniques developed are then showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids, and rough path theory. With plentiful exercises, some with solutions, and worked examples, this will be indispensable for graduate students and researchers working at the intersection of functional analysis, non-linear differential equations and differential geometry. This title is also available as Open Access on Cambridge Core.
Contains over 180 exercises, many with solutions
Showcases connections between finite and infinite-dimensional geometry
Introduces relevant modern applications and active research areas such as shape analysis, geometric hydrodynamics and rough path theory
This book is also available as Open Access on Cambridge Core
1. Calculus in locally convex spaces
2. Spaces and manifolds of smooth maps
3. Lifting geometry to mapping spaces I: Lie groups
4. Lifting geometry to mapping spaces II: (weak) Riemannian metrics
5. Weak Riemannian metrics with applications in shape analysis
6. Connecting finite-dimensional, infinite-dimensional and higher geometry
7. Euler?Arnold theory: PDE via geometry
8. The geometry of rough paths
A. A primer on topological vector spaces and locally convex spaces
B. Basic ideas from topology
C. Canonical manifold of mappings
D. Vector fields and their Lie bracket
E. Differential forms on infinite-dimensional manifolds
F. Solutions to selected exercises
References
Index.
Part of New Mathematical Monographs
Not yet published - available from October 2022
FORMAT: Hardback ISBN: 9781108832038
Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.
Lays a rigorous groundwork in dealing with arithmetic groups in algebraic groups
Includes worked examples scattered throughout the text, as well as open ends for further research
Unfolds the essential geometric and number theoretical components to the investigations of arithmetic groups
Part I. Arithmetic Groups in the General Linear Group:
1. Modules, lattices, and orders
2. The general linear group over rings
3. A menagerie of examples ? a historical perspective
4. Arithmetic groups
5. Arithmetically defined Kleinian groups and hyperbolic 3-space
Part II. Arithmetic Groups Over Global Fields:
6. Lattices ? Reduction theory for GLn
7. Reduction theory and (semi)-stable lattices
8. Arithmetic groups in algebraic k-groups
9. Arithmetic groups, ambient Lie groups, and related geometric objects
10. Geometric cycles
11. Geometric cycles via rational automorphisms
12. Reduction theory for adelic coset spaces
Appendices: A. Linear algebraic groups ? a review
B. Global fields
C. Topological groups, homogeneous spaces, and proper actions
References
Index.