TEXTBOOK
Part of Cambridge Mathematical Textbooks
PUBLICATION PLANNED FOR: April 2019
HardbackISBN: 9781107134829
This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass' approach, stressing the importance of power series expansions instead of starting with the Cauchy integral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville's theorem, and Schwarz's lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics.
Advance praise: 'Marshall's book covers the basic topics with crystal clarity in a style that is conversational and concrete, and that guides the student into thinking about these topics the way a working mathematician does, especially one with a geometric or computational bent. Moreover, the book includes many results that are vital to modern function theory and its applications to geometry, dynamics and probability, but that are often omitted from introductory texts. I wish I had first learned the subject from this book, and I am delighted that my students can do so.' Christopher Bishop, State University of New York at Stony Brook
PUBLICATION PLANNED FOR: March 2019
Hardback ISBN: 9781108424936
Paperback ISBN: 9781108441025
Differential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the Gauss?Bonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercises to test students' understanding of the material, and ends with a supplementary chapter on minimal surfaces that could be used as an extension towards advanced courses or as a source of student projects.
Preface
1. Curves in Rn
2. Surfaces in Rn
3. Smooth maps
4. Measuring how surfaces curve
5. The Theorema Egregium
6. Geodesic curvature and geodesics
7. The Gauss-Bonnet theorem
8. Minimal and CMC surfaces
9. Hints of answers to some exercises
Index.
Part of Cambridge Tracts in Mathematics
PUBLICATION PLANNED FOR: December 2018
HardbackISBN: 9781108474429
Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.
Changes the field from a collection of disparate results into a coherent body of knowledge
Over 350 exercises help the reader gain a deeper understanding of the concepts
Presents more than 100 open questions suitable for dissertation topics or further research
Introduction
1. Topological rings and modules and their completions
2. Inverse limits
3. The idea of slenderness
4. Objects of type ? / \coprod
5. Concrete examples. Slender rings
6. More examples of slender objects
Appendix. Ordered sets and measurable cardinals
References
Notation index
Name Index
Subject index.
Part of Encyclopedia of Mathematics and its Applications
PUBLICATION PLANNED FOR: April 2019
HardbackISBN: 9781108416849
Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject.
Provides a unified perspective, allowing readers to see the big picture rather than only their specific area
Covers all the essentials so that newcomers can quickly get up to speed
Describes how various ideas manifest in different areas of the field, making clear the connections between them
Part of New Mathematical Monographs
PUBLICATION PLANNED FOR: May 2019
HardbackISBN: 9781107146723
Spectral spaces are a class of topological spaces. They are a tool linking algebraic structures, in a very wide sense, with geometry. They were invented to give a functional representation of Boolean algebras and distributive lattices and subsequently gained great prominence as a consequence of Grothendieck's invention of schemes. There are more than 1000 research articles about spectral spaces, but this is the first monograph. It provides an introduction to the subject and is a unified treatment of results scattered across the literature, filling in gaps and showing the connections between different results. The book also includes new research going beyond the existing literature, answering questions that naturally arise from this comprehensive approach. The authors serve graduates by starting gently with the basics. For experts, they lead them to the frontiers of current research, making this book a valuable reference source.
Presents many applications of spectral spaces, their benefits, and how they naturally arise in different contexts
Contains a large number of examples and counterexamples to help the reader learn the material
Comprehensive indexes make the book a useful reference resource
Part of Encyclopedia of Mathematics and its Applications
PUBLICATION PLANNED FOR: April 2019
HardbackISBN: 9781108427012
This is the first book to be dedicated entirely to Drinfeld's quasi-Hopf algebras. Ideal for graduate students and researchers in mathematics and mathematical physics, this treatment is largely self-contained, taking the reader from the basics, with complete proofs, to much more advanced topics, with almost complete proofs. Many of the proofs are based on general categorical results; the same approach can then be used in the study of other Hopf-type algebras, for example Turaev or Zunino Hopf algebras, Hom-Hopf algebras, Hopfish algebras, and in general any algebra for which the category of representations is monoidal. Newcomers to the subject will appreciate the detailed introduction to (braided) monoidal categories, (co)algebras and the other tools they will need in this area. More advanced readers will benefit from having recent research gathered in one place, with open questions to inspire their own research.
Introduces beginners to the basics of quasi-Hopf algebras, including categorical machinery necessary for their study
Contains open problems which give the reader inspiration for future research
Brings together several advanced topics for the first time in one book
1. Monoidal and braided categories
2. Algebras and coalgebras in monoidal categories
3. Quasi-bialgebras and quasi-Hopf algebras
4. Module (co)algebras and (bi)comodule algebras
5. Crossed products
6. Quasi-Hopf bimodule categories
7. Finite-dimensional quasi-Hopf algebras
8. Yetter?Drinfeld module categories
9. Two-sided two-cosided Hopf modules
10. Quasitriangular quasi-Hopf algebras
11. Factorizable quasi-Hopf algebras
12. The quantum dimension and involutory quasi-Hopf algebras
13. Ribbon quasi-Hopf algebras
Bibliography
Index.