Bücher der Reihe Graduate Texts in Mathematics

It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

This book is an introduction to manifolds at the beginning graduate level, and accessible to any student who has completed a solid undergraduate degree in mathematics.

This textbook introduces the representation theory of algebras by focusing on two of its most important aspects: the Auslander-Reiten theory and the study of the radical of a module category. It then turns to the study of endomorphism algebras, leading on one hand to the definition of the Auslander algebra and on the other to tilting theory.

The fourth edition of this standard textbook of modern graph theory has been revised, updated, and substantially extended. Covering all major recent developments, it can be used both as a reliable textbook for an introductory course and as a graduate text.

Based on the lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Munster, this book provides a modern introduction to this subject, presenting methods used in the study of complex manifolds in the special case of complex dimension one.

This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics.

This updated third edition of Shiryaev's work on probability contains a systematic treatment from the ground up, starting with intuitive ideas, then developing more sophisticated subjects. Examples are discussed in detail, and there are a large number of exercises.

Addresses many problems on the theory of functions of one complex variable. This book treats several topics in geometric function theory and potential theory in the plane, such as conformal equivalence for simply connected regions, de Branges' proof of the Bieberbach conjecture, Hardy spaces on the disk, and potential theory in the plane.

The theory of elliptic curves is distinguished by the diversity of the methods used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry.

This text is a rigorous introduction to Ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. It describes some recent applications to number theory, and goes beyond the standard texts in this topic.

This text offers a thorough, modern introduction to commutative algebra. It concentrates on concepts and results at the center of the field while keeping a constant view on the natural geometrical context. It includes many examples and exercises.

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves;

This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P.

This book is intended as a basic text for a one year course in algebra at the graduate level or as a useful reference for mathematicians and professionals who use higher-level algebra.

This is an expanded edition of a popular textbook that provides a purely algebraic, self-contained and in-depth exposition of the theory of function fields. It contains numerous exercises, some fairly simple, some quite difficult.

"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis.

This is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory.

The primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e.

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds.

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory.

This book is meant as a text for a first-year graduate course in analysis. This allows a course to omit material from some chapters without compromising the exposition of material from later chapters.

The book concludes with two supplementary chapters: a concise account of unbounded operators and their spectral theory, and a complete course in measure and integration theory from an advanced point of view.

This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted."

Some of the central topics in number theory, presnted in a simple and concise fashion. All the topics are presented in a refreshingly elegant and efficient manner with clever examples and interesting problems throughout. The text is suitable for a graduate course in analytic number theory.

Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises.

Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

A discussion of certain advanced topics in operator theory, providing the necessary background while assuming only standard senior-first year graduate courses in general topology, measure theory, and algebra. This new edition will appeal to a whole new generation of students seeking an introduction to this topic.