Global Calculus (Graduate Studies in Mathematics, 65)

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 scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on global analysis. The Power That Analysis, Topology And Algebra Bring To Geometry Has Revolutionised 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 Scratch. The Basics Of Fourier Transforms, Sobolev Theory And Interior Regularity Are Proved At The Same Time As Symbol Calculus, Culminating In Beautiful Results In Global Analysis, Real And Complex. Many New Perspectives On Traditional And Modern Questions Of Differential Analysis And Geometry Are The Hallmarks Of The Book. The Book Is Suitable For A First Year Graduate Course On Global Analysis.--book Jacket. Ch. 1. Sheaves And Differential Manifolds : Definitions And Examples -- Ch. 2. Differential Operators -- Ch. 3. Integration On Differential Manifolds -- Ch. 4. Cohomology Of Sheaves And Applications -- Ch. 5. Connections On Principal And Vector Bundles; Lifting Of Symbols -- Ch. 6. Linear Connections -- Ch. 7. Manifolds With Additional Structures -- Ch. 8. Local Analysis Of Elliptic Operators -- Ch. 9. Vanishing Theorems And Applications -- App. 1. Algebra -- App. 2. Topology -- App. 3. Analysis. S. Ramanan. Includes Bibliographical References (p. 311-312) And Index. Ramanan (U. of California at Los Angeles) writes primarily for first-year graduate students interested in differential operations. Unlike most texts, this one begins by describing sheaves before differential manifolds, and then continues to an examination of connective algebra. Ramanan also takes a nontraditional approach to densities and orientations in which the change-of-variable formula is only a consequence. He follows this with descriptions sheaf cohomology connections, additional structures, the local theory of elliptic operators and finally the composition of vanishing theorems. Ramanan includes exercised throughout. Annotation 2004 Book News, Inc., Portland, OR (booknews.com)