Presents a cross-section of different aspects of Riemann surfaces, covering the basics as well as highlighting new developments in the field. Written by leading researchers, it will be suitable for gr
This book provides a careful and accessible exposition of the function analytic approach to initial boundary value problems for semilinear parabolic differential equations. This book will be a necessa
This book falls into two parts. The first studies those circle diffeomorphisms which commute with a diffeomorphism f whose rotation number r(f) is irrational. When f is not smoothly conjugated to a ro
Perturbation of the boundary is a rather neglected topic in the study of PDEs for two main reasons. First, on the surface it appears trivial, merely a change of variables and an application of the cha
Integral closure has played a role in number theory and algebraic geometry since the nineteenth century, but a modern formulation of the concept for ideals perhaps began with the work of Krull and Zariski in the 1930s. It has developed into a tool for the analysis of many algebraic and geometric problems. This book collects together the central notions of integral closure and presents a unified treatment. Techniques and topics covered include: behavior of the Noetherian property under integral closure, analytically unramified rings, the conductor, field separability, valuations, Rees algebras, Rees valuations, reductions, multiplicity, mixed multiplicity, joint reductions, the Briançon-Skoda theorem, Zariski's theory of integrally closed ideals in two-dimensional regular local rings, computational aspects, adjoints of ideals and normal homomorphisms. With many worked examples and exercises, this book will provide graduate students and researchers in commutative algebra or ring theory w
This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research.
The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story.
Written for mathematicians working with the theory of graph spectra, this book explores more than 400 inequalities for eigenvalues of the six matrices associated with finite simple graphs: the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, Seidel matrix, and distance matrix. The book begins with a brief survey of the main results and selected applications to related topics, including chemistry, physics, biology, computer science, and control theory. The author then proceeds to detail proofs, discussions, comparisons, examples, and exercises. Each chapter ends with a brief survey of further results. The author also points to open problems and gives ideas for further reading.
This text provides a complete introduction to the theory of variational inequalities with emphasis on contact mechanics. It covers existence, uniqueness and convergence results for variational inequalities, including the modelling and variational analysis of specific frictional contact problems with elastic, viscoelastic and viscoplastic materials. New models of contact are presented, including contact of piezoelectric materials. Particular attention is paid to the study of history-dependent quasivariational inequalities and to their applications in the study of contact problems with unilateral constraints. The book fully illustrates the cross-fertilisation between modelling and applications on the one hand and nonlinear mathematical analysis on the other. Indeed, the reader will gain an understanding of how new and nonstandard models in contact mechanics lead to new types of variational inequalities and, conversely, how abstract results concerning variational inequalities can be
D-module theory is essentially the algebraic study of systems of linear partial differential equations. This book, the first devoted specifically to holonomic D-modules, provides a unified treatment of both regular and irregular D-modules. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on D-modules and their tempered holomorphic solutions. As an application, they obtain the Riemann–Hilbert correspondence for regular holonomic D-modules. In the second part of the book the authors do the same for the sheaf of enhanced tempered solutions of (not necessarily regular) holonomic D-modules. Originating from a series of lectures given at the Institut des Hautes Études Scientifiques in Paris, this book is addressed to graduate students and researchers familiar with the language of sheaves and D-modules, in the derived sense.
This in-depth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs. Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. The final four chapters are devoted to the reconstruction problem, and here special emphasis is given to those results that involve the symmetry of graphs, many of which are not to be found in other books. This second edition expands on several of the topics found in the first edition and includes both an enriched bibliography and a wide collection of exercises. Clearer proofs are provided, as are new examples of graphs with interesting symmetry properties. Any student who masters the contents of this book will be well prepared for current research in many aspects of the theory of graph automorphisms and the reconstruction problem.
This volume takes its name from a popular series of intensive mathematics workshops hosted at institutions in Appalachia and surrounding areas. At these meetings, internationally prominent set theorists give one-day lectures that focus on important new directions, methods, tools and results so that non-experts can begin to master these and incorporate them into their own research. Each chapter in this volume was written by the workshop leaders in collaboration with select student participants, and together they represent most of the meetings from the period 2006–2012. Topics covered include forcing and large cardinals, descriptive set theory, and applications of set theoretic ideas in group theory and analysis, making this volume essential reading for a wide range of researchers and graduate students.
Number theory currently has at least three different perspectives on non-abelian phenomena: the Langlands programme, non-commutative Iwasawa theory and anabelian geometry. In the second half of 2009, experts from each of these three areas gathered at the Isaac Newton Institute in Cambridge to explain the latest advances in their research and to investigate possible avenues of future investigation and collaboration. For those in attendance, the overwhelming impression was that number theory is going through a tumultuous period of theory-building and experimentation analogous to the late 19th century, when many different special reciprocity laws of abelian class field theory were formulated before knowledge of the Artin–Takagi theory. Non-abelian Fundamental Groups and Iwasawa Theory presents the state of the art in theorems, conjectures and speculations that point the way towards a new synthesis, an as-yet-undiscovered unified theory of non-abelian arithmetic geometry.
