The power that analysis, topology and algebra bring to geometry has revolutionized the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from...

This is the second volume of a two-volume graduate text in set theory. The first volume covered the basics of modern set theory and was addressed primarily to beginning graduate students. This second volume is intended as a bridge between introductory set theory courses such as the first volume and advanced monographs that cover selected branches of set theory. The authors give short but rigorous introductions to set-theoretic concepts and techniques such as trees, partition calculus, cardinal invariants of the continuum, Martin's Axiom, closed unbounded and stationary sets, the Diamond...

This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. It explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. It deals with vector fields and differential forms, addresses integration of vector fields and p-planes and develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant...

This work treats the fundamentals of differential geometry - manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry.

Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory.

Presents two essential and apparently unrelated subjects that include, microlocal analysis and the theory of pseudo-differential operators, a basic tool in the study of partial differential equations and in analysis on manifolds; and the Nash-Moser theorem

Presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds. This book unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.

Defines simplicial homology and cohomology, with various examples and applications. This book introduces Kolmogorov-Alexander multiplication in cohomology. It deals with the applications of simplicial homology and cohomology to obstruction theory, in parti

This is an elementary, self-contained presentation of the integration processes developed by Lebesgue, Denjoy, Perron, and Henstock. The Lebesgue integral and its essential properties are first developed in detail. The other three integrals are all generalizations of the Lebesgue integral that satisfy the ideal version of the Fundamental Theorem of Calculus: if F is differentiable on the interval a, b], then F ]1 integrable on a, b] and f ]b a = F(b) - F(a). One of the book's unique features is that the Denjoy, Perron, and Henstock integrals are each developed carefully from their...

A companion volume to 'Graduate Algebra - Commutative View', the most important feature of this book is that it presents a unified approach to many important topics, such as group theory, ring theory, Lie algebras, and gives conceptual proofs of many basic results of noncommutative algebra.