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This volume, derived from the 'PDEs in Fluid Mechanics' workshop held at the University of Warwick in 2016, serves to consolidate and advance work in mathematical fluid dynamics. Consisting of surveys and original research, it will be a valuable resource for both established researchers and graduate students seeking an overview of current developments.

The stability of stochastic differential equations in abstract, mainly Hilbert, spaces receives a unified treatment in this self-contained book. It covers basic theory as well as computational techniques. It will be useful for researchers across numerical computation, engineering, and mathematical physics and biology.

This book focuses on generalisations of Gromov hyperbolicity in geometric group theory. Five self-contained expository articles introduce topics 'beyond hyperbolicity': these can be used as an introduction for students or as a reference for experts. The final part contains research articles on the latest results in this rich and active field.

The eight articles in this book provide a valuable survey of the present state of knowledge in combinatorics. Written by leading experts in the field, they provide expanded accounts of plenary seminars given at the British Combinatorial Conference in July 2019.

Created as a celebration of mathematical pioneer Emma Previato, this comprehensive second volume highlights the connections between her main fields of research, namely algebraic geometry and integrable systems. Written by leaders in the field, the text is accessible to graduate students and non-experts, as well as researchers.

The book is designed for graduate students and beginning researchers into the arithmetic theory of automorphic forms, and for all who want to know more about the Langlands program. It forms a sequel to On the Stabilization of the Trace Formula published in 2011.

Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between her main fields of research, namely algebraic geometry and integrable systems. Written by leaders in the field, the text is accessible to graduate students and non-experts, as well as researchers.

This book addresses the interplay between several rapidly expanding areas of mathematics. Suitable for graduate students as well as researchers, it provides surveys of topics linking geometry, spectral theory and stochastics.

This monograph presents the transformation theory of Willmore surfaces (possibly with a constraint), including applications to surfaces of constant mean curvature. Self-contained accounts of the core topics make it suitable for newcomers to the field, while many detailed computations and new results make it an appealing reference for experts.

Wigner's theorem plays an important role in the mathematical foundations of quantum mechanics. This book provides a quick, accessible introduction to the geometric approach to Wigner-type theorems, unifying both classical and more recent results, and is suitable for graduate students as well as more experienced researchers.

This is the first book on the topic of study of modules invariant under automorphisms of their envelopes and covers. Containing plentiful examples and open problems, it is a valuable resource for graduate students and researchers in algebra who wish to learn the state of the art in this area of module theory.

Zeta and L-functions have played a major part in the development of number theory. This book for graduate students and researchers presents a big picture of some key results and surrounding theory, whilst taking the reader on a journey through the history of their development.

This book covers the various hard mathematical problems used in cryptography and on which cybersecurity is built, as well as recent advances on how to solve these problems from a theoretic and practical applied perspective. Students and security engineers will benefit from these leading experts' overviews of computational cryptography.

The nine survey articles in this book provide expanded accounts of plenary lectures given at the British Combinatorial Conference in July 2021. Written by leading experts in the field, they present the state of the art in several areas of contemporary interest in combinatorics.

Robert Langlands, recipient of the Abel Prize, profoundly changed the mathematical landscape with his creation of the Langlands Program. This book, designed to serve a wide range of mathematicians and advanced students, guides readers through the technically difficult areas of Langlands' work and presents a step-by-step account of his process.

This volume arises from the 2019 'Equivariant Topology and Derived Algebra' conference (Trondheim). The papers, written by experts in the field, have been carefully selected to contain a balance of new research and expository papers, accessible to new researchers in the area and forming a valuable reference for experts.

This is the first book on matrix and kernel transforms preserving positivity structures. It is self-contained and only requires modest prerequisites in analysis and linear algebra. Covering deep results and modern progress, the book succeeds as an introduction for beginners and a comprehensive reference for experts.

This modern introduction to operator theory on spaces with indefinite inner product guides the reader step by step through the important results of recent decades. It is highly readable and contains generous background material, making it suitable for graduate students and researchers in operator theory, functional analysis, and related areas.

This book provides a comprehensive guide to Diophantine equations over finitely generated domains, with a focus on proving effective finiteness results. No specialized knowledge is required, enabling graduate students and experts alike to learn the necessary techniques and apply them in their own research.

Written to honor the 80th birthday of William Fulton, the articles in this volume present substantial contributions to algebraic geometry, particularly combinatorial algebraic geometry and intersection theory. Covering a wide range of topics of current interest, the book will appeal to graduate students and established researchers in the field.

Written to honor the 80th birthday of William Fulton, the articles in this volume present substantial contributions to algebraic geometry, particularly combinatorial algebraic geometry and intersection theory. Covering a wide range of topics of current interest, the book will appeal to graduate students and established researchers in the field.

The purpose of this book is to provide an introduction to the theory of jet bundles for mathematicians and physicists who wish to study differential equations, particularly those associated with the calculus of variations, in a modern geometric way. One of the themes of the book is that first-order jets may be considered as the natural generalisation of vector fields for studying variational problems in field theory, and so many of the constructions are introduced in the context of first- or second-order jets, before being described in their full generality. The book includes a proof of the local exactness of the variational bicomplex. A knowledge of differential geometry is assumed by the author, although introductory chapters include the necessary background of fibred manifolds, and on vector and affine bundles. Coordinate-free techniques are used throughout, although coordinate representations are often used in proofs and when considering applications.

Sheaf theory provides a means of discussing many different kinds of geometric objects in respect of the connection between their local and global properties. It finds its main applications in topology and modern algebraic geometry where it has been used as a tool for solving, with great success, several long-standing problems. This text is based on a lecture course for graduate pure mathematicians which builds up enough of the foundations of sheaf theory to give a broad definition of manifold, covering as special cases the algebraic geometer's schemes as well as the topological, differentiable and analytic kinds, and to define sheaf cohomology for application to such objects. Exercises are provided at the end of each chapter and at various places in the text. Hints and solutions to some of them are given at the end of the book.

A fundamental problem of algebraic topology is the classification of homotopy types and homotopy classes of maps. In this work the author extends results of rational homotopy theory to a subring of the rationale. The methods of proof employ classical commutator calculus of nilpotent group and Lie algebra theory and rely on an extensive and systematic study of the algebraic properties of the classical homotopy operations (composition and addition of maps, smash products, Whitehead products and higher order James-Hopi invariants). The account is essentially self-contained and should be accessible to non-specialists and graduate students with some background in algebraic topology and homotopy theory.

A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic curves. This approach provides geometers with the opportunity to acquaint themselves with aspects of their subject virtually ignored by other texts. The exposition is clear and logically carries themes from earlier through to later topics. This enthusiastic work of scholarship is made complete with the inclusion of some interesting historical details and a very comprehensive bibliography.

The most important examples of finite groups are the group of permutations of a set of n objects, known as the symmetric group, and the group of non-singular n-by-n matrices over a finite field, which is called the general linear group. This book examines the representation theory of the general linear groups, and reveals that there is a close analogy with that of the symmetric groups. It consists of an essay which was joint winner of the Cambridge University Adams Prize 1981-2, and is intended to be accessible to mathematicians with no previous specialist knowledge of the topics involved. Many people have studied the representations of general linear groups over fields of the natural characteristic, but this volume explores new territory by considering the case where the characteristic of the ground field is not the natural one. Not only are the results in the book elegant and interesting in their own right, but they suggest many lines for further investigation.

This account is a study of twofold symmetry in algebraic topology. The author discusses specifically the antipodal involution of a real vector bundle - multiplication by - I in each fibre; doubling and squaring operations; the symmetry of bilinear forms and Hermitian K-theory. In spite of its title, this is not a treatise on equivariant topology; rather it is the language in which to describe the symmetry. Familiarity with the basic concepts of algebraic topology (homotopy, stable homotopy, homology, K-theory, the Pontrjagin-Thom transfer construction) is assumed. Detailed proofs are not given (the expert reader will be able to supply them when necessary) yet nowhere is credibility lost. Thus the approach is elementary enough to provide an introduction to the subject suitable for graduate students although research workers will find here much of interest.

This book considers convergence of adapted sequences of real and Banach space-valued integrable functions, emphasizing the use of stopping time techniques. Not only are highly specialized results given, but also elementary applications of these results. The book starts by discussing the convergence theory of martingales and sub-( or super-) martingales with values in a Banach space with or without the Radon-Nikodym property. Several inequalities which are of use in the study of the convergence of more general adapted sequence such as (uniform) amarts, mils and pramarts are proved and sub- and superpramarts are discussed and applied to the convergence of pramarts. Most of the results have a strong relationship with (or in fact are characterizations of) topological or geometrical properties of Banach spaces. The book will interest research and graduate students in probability theory, functional analysis and measure theory, as well as proving a useful textbook for specialized courses on martingale theory.

This volume contains selected contributions from a very successful meeting on Number Theory and Dynamical Systems held at the University of York in 1987. There are close and surprising connections between number theory and dynamical systems. One emerged last century from the study of the stability of the solar system where problems of small divisors associated with the near resonance of planetary frequencies arose. Previously the question of the stability of the solar system was answered in more general terms by the celebrated KAM theorem, in which the relationship between near resonance (and so Diophantine approximation) and stability is of central importance. Other examples of the connections involve the work of Szemeredi and Furstenberg, and Sprindzuk. As well as containing results on the relationship between number theory and dynamical systems, the book also includes some more speculative and exploratory work which should stimulate interest in different approaches to old problems.

Design theory is a branch of combinatorics with applications in number theory, coding theory and geometry. In this book the authors discuss the generalization of results and applications to quasi-symmetric designs. The coverage is comprehensive and will be useful for researchers and graduate students. An attractive feature is the discussion of unsolved problems.