Part of Cambridge Mathematical Textbooks
Publication planned for: August 2018
availability: Not yet published - available from August 2018
format: Hardback
isbn: 9781108425797
Manifolds are abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of local-to-global properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is hands-on, including many concrete examples and exercises woven into the text, with hints provided to guide the student.
Advance praise: 'Learning some topics in mathematics is a bit like climbing a mountain - it is best done with a guide. In this short course, Dundas is just that guide - revealing the best routes, giving the reader first-hand experience through lots of well-chosen exercises, providing relevant and motivating examples, and finally, making it all fun.' John McCleary, Vassar College, New York
1. Introduction
2. Smooth manifolds
3. The tangent space
4. Regular values
5. Vector bundles
6. Constructions on vector bundles
7. Integrability
8. Local phenomena that go global
Appendix A. Point set topology
Appendix B. Hints or solutions to exercises
References
Index.
Publication planned for: September 2018
availability: Not yet published - available from September 2018
format: Paperback
isbn: 9781108436793
This new edition of a classic textbook develops complex analysis from the established theory of real analysis by emphasising the differences that arise as a result of the richer geometry of the complex plane. Key features of the authors' approach are to use simple topological ideas to translate visual intuition to rigorous proof, and, in this edition, to address the conceptual conflicts between pure and applied approaches head-on. Beyond the material of the clarified and corrected original edition, there are three new chapters: Chapter 15 on infinitesimals in real and complex analysis; Chapter 16 on homology versions of Cauchy's Theorem and Cauchy's Residue Theorem, linking back to geometric intuition; and Chapter 17 outlines some more advanced directions in which complex analysis has developed, and continues to evolve into the future. With numerous worked examples and exercises, clear and direct proofs, and a view to the future of the subject, this is an invaluable companion for any modern complex analysis course.
Using simple topological ideas of continuity and connectivity, this textbook explains the differences between real and complex analysis as a consequence of the richer geometry of the complex plane and teaches students to translate visual intuition into rigorous proof.
Introduces a simple formal definition of an extension field containing infinitesimal quantities to show the connection between pure and applied approaches ? students grasp the continuing evolution of mathematical ideas.
Supplementary material including a concordance showing in more detail the changes between the previous edition and this one, and links to GeoGebra worksheets illustrating the ideas of chapter 15, can be found on the Cambridge University Press website: http://www.cambridge.org
Preface to the First Edition
Preface to the Second Edition
0. The Origins of Complex Analysis, and Its Challenge to Intuition
1. Algebra of the Complex Plane
2. Topology of the Complex Plane
3. Power Series
4. Diff erentiation
5. The Exponential Function
6. Integration
7. Angles, Logarithms, and the Winding Number
8. Cauchy's Theorem
9. Homotopy Versions of Cauchy's Theorem
10. Taylor Series
11. Laurent Series
12. Residues
13. Conformal Transformations
14. Analytic Continuation
15. Infinitesimals in Real and Complex Analysis
16. Homology Version of Cauchy's Theorem
17. The Road Goes Ever On
References
Index
Part of London Mathematical Society Lecture Note Series
availability: Not yet published - available from September 2018
format: Paperback
isbn: 9781108437622
This volume collects the proceedings of the conference “Topological methods in group theory”, held at Ohio State University in 2014 in honor of Ross Geoghegan's 70th birthday. It consists of eleven peer-reviewed papers on some of the most recent developments at the interface of topology and geometric group theory. The authors have given particular attention to clear exposition, making this volume especially useful for graduate students and for mathematicians in other areas interested in gaining a taste of this rich and active field. A wide cross-section of topics in geometric group theory and topology are represented, including left-orderable groups, groups de?ned by automata, connectivity properties and Σ-invariants of groups, amenability and non-amenability problems, and boundaries of certain groups. Also included are topics that are more geometric or topological in nature, such as the geometry of simplices, decomposition complexity of certain groups, and problems in shape theory.
Brings the reader right up to date with the latest developments at the interface between geometric group theory and topology
Contains eleven fully peer-reviewed papers
Careful exposition throughout makes this volume particularly useful for graduate students and mathematicians working in other areas
List of contributors
1. Left relatively convex subgroups Yago Antolin, Warren Dicks, and Zoran ?uni?
2. Groups with context-free co-word problem and embeddings into Thompson's group V Rose Berns-Zieve, Dana Fry, Johnny Gillings, Hannah Hoganson, and Heather Mathews
3. Limit sets for modules over groups acting on a CAT(0) space Robert Bieri and Ross Geoghegan
4. Ideal structure of the C*-algebra of R. Thompson's group T Collin Bleak and Kate Juschenko
5. Local similarity groups with context-free co-word problem Daniel Farley
6. Compacta with shapes of finite complexes: a direct approach to the Edwards-Geoghegan-Wall obstruction Craig R. Guilbault
7. The horofunction boundary of the lamplighter group L2 with the Diestel-Leader metric Keith Jones and Gregory A. Kelsey
8. Intrinsic geometry of a Euclidean simplex Barry Minemyer
9. Hyperbolic dimension and decomposition complexity Andrew Nicas and David Rosenthal
10. Some remarks on the covering groups of a topological group Dongwen Qi
11. The Σ-invariants of Thompson's group F via Morse Theory Stefan Witzel and Matthew C. B. Zaremsky.
Part of Cambridge Studies in Advanced Mathematics
availability: Not yet published - available from October 2018
format: Hardback
isbn: 9781107187733
This graduate textbook offers a self-contained introduction to the concepts and techniques of logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry and number theory. It features a systematic exposition of the foundations of the field, from the basic results on convex geometry and commutative monoids to the theory of logarithmic schemes and their de Rham and Betti cohomology. The book will be of use to graduate students and researchers working in algebraic, analytic, and arithmetic geometry as well as related fields.
Brings together numerous results across the field into a single, comprehensive reference
This book is accessible to readers from a wide variety of backgrounds
Includes detailed proofs and careful definitions of the main results and concepts
1. The geometry of monoids
2. Sheaves of monoids
3. Logarithmic schemes
4. Differentials and smoothness
5. Betti and de Rham cohomology.
Hardback
May 1, 2018 Forthcoming
Reference - 236 Pages
ISBN 9781138585997
Series: Chapman & Hall/CRC Monographs on Statistics & Applied Probability
Accessible to practitioners and researchers with a wide range of backgrounds and interests in network science
Explains the logical considerations of network modeling needed to evaluate the suitability of existing models and
develop new models as appropriate
Distills technical considerations of network modeling in a manner accessible to both industry practitioners and
academic researchers
Thought provoking questions focus the reader's attention as well as foster class discussion and inspire new
research directions.
Probabilistic Foundations of Statistical Network Analysis presents a fresh and insightful perspective on the fundamental tenets and major challenges of modern network analysis. Its lucid exposition provides necessary background for understanding the essential ideas behind exchangeable and dynamic network models, network sampling, and network statistics such as sparsity and power law, all of which play a central role in contemporary data science and machine learning applications. The book rewards readers with a clear and intuitive understanding of the subtle interplay between basic principles of statistical inference, empirical properties of network data, and technical concepts from probability theory. Its mathematically rigorous, yet non-technical, exposition makes the book accessible to professional data scientists, statisticians, and computer scientists as well as practitioners and researchers in substantive fields. Newcomers and non-quantitative researchers will find its conceptual approach invaluable for developing intuition about technical ideas from statistics and probability, while experts and graduate students will find the book a handy reference for a wide range of new topics, including edge exchangeability, relative exchangeability, graphon and graphex models, and graph-valued Levy process and rewiring models for dynamic networks.
The author’s incisive commentary supplements these core concepts, challenging the reader to push beyond the current limitations of this emerging discipline. With an approachable exposition and more than 50 open research problems and exercises with solutions, this book is ideal for advanced undergraduate and graduate students interested in modern network analysis, data science, machine learning, and statistics.