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Introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields.

This book provides an introduction to quantum theory primarily for mathematics students, and is based on the sort of material usually covered in the first two years of undergraduate algebra. It includes more mathematical detail than is common in introductory books at this level and covers some more advanced topics.

This volume presents an array of topics that introduce the reader to key ideas in active areas in geometry and topology. The topics covered range from Morse theory and complex geometry to geometric group theory, and the text is accompanied by exercises that are designed to deepen the reader's understanding.

Bundles, connections, metrics & curvature are the lingua franca of modern differential geometry & theoretical physics. Supplying graduate students in mathematics or theoretical physics with the fundamentals of these objects, & providing numerous examples, the book would suit a one-semester course on the subject of bundles & the associated geometry

Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. The book is aimed at graduate students of mathematics and related disciplines wishing to learn the basics and techniques of Real Analysis with reasonable ease.

A text aimed at third year undergraduates and graduates in mathematics and physics, presenting elementary twistor theory as a universal technique for solving differential equations in applied mathematics and theoretical physics.

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The authors are internationally renowned both as researchers and expositors, and the book is written in an informal and accessible style.

Stochastic filtering theory is a field that has seen a rapid development in recent years and this book, aimed at graduates and researchers in applied mathematics, provides an accessible introduction covering recent developments.

This text focuses on the extraordinary success of quantum cohomology and its connections with many existing areas of traditional mathematics and new areas such as mirror symmetry. Aimed at graduate students in mathematics as well as theoretical physicists, the text assumes basic familiarity with differential equations and cohomology.

A text that will bring together PDE theory, general relativity and astrophysics to deliver an overview of theory of partial differential equations for general relativity. The text will include numerous examples and provide a unique resource for graduate students in mathematics and physics, numerical relativity and cosmology.

Aimed at graduates with a grounding in complex analysis, this book provides an accessible introduction to the theory of quasiconformal maps and Teichmuller theory. Assuming some prior familiarity with Riemann surfaces and hyperbolic geometry, the text is illustrated throughout by examples and exercises.

This introduction to topology emphasises a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style making the student an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students.

An introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level, this work covers the essentials of the subject. Starting from basic undergraduate mathematics, it proceeds through the fundamentals of Lie theory up to topics in representation theory.

This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

Topology and invariants of smooth 4-manifolds are presented in a self-contained way. In particular handlebodies of 4-manifolds, symplectic and stein structures, and seiberg-witten invariants are discussed.

Beginning with the concept of random processes and Brownian motion and building on the theory and research directions in a self-contained manner, this book provides an introduction to stochastic analysis for graduate students, researchers and applied scientists interested in stochastic processes and their applications.

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

An authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools.

A timely graduate level text in an active field covering functional analysis, with an emphasis on Banach algebras.

Riemannian holonomy groups and calibrated geometry covers an exciting and active area of research at the crossroads of several different fields in Mathematics and Physics. Drawing on the author's previous work the text has been written to explain the advanced mathematics involved simply and clearly to graduate students in both disciplines.

A text aimed at both geometers needing the tools of rational homotopy theory to understand and discover new results concerning various geometric subjects, and topologists who require greater breadth of knowledge about geometric applications of the algebra of homotopy theory.

This text provides a modern introduction to a central part of mathematical analysis. It can be used as a self-contained textbook for beginner courses in functional analysis. In its last chapter recent results from the theory of Frechet spaces are presented.

Categories for Quantum Theory: An Introduction lays foundations for an approach to quantum theory that uses category theory, a branch of pure mathematics. Prior knowledge of quantum information theory or category theory helps, but is not assumed, and basic linear algebra and group theory suffices.

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. This new third edition of a classic book in the feild includes updates and new material to bring the material right up-to-date.

This major revision of James Oxley's classic Matroid Theory provides a comprehensive introduction to the subject, covering the basics to more advanced topics. With over 700 exercises and proofs of all relevant major theorems, this book is the ideal reference and class text for academics and graduate students in mathematics and computer science.

This graduate text shows how the computer can be used as a tool for research in number theory through numerical experimentation. Examples of experiments in binary quadratic forms, zeta functions of varieties over finite fields, elementary class field theory, elliptic units, modular forms, are provided along with exercises and selected solutions.

This graduate level text covers the theory of stochastic integration, an important area of Mathematics with a wide range of applications, including financial mathematics and signal processing.