Bücher der Reihe Oxford Graduate Texts in Mathematics

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.

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.

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.

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.

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 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.

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.