Series: London Mathematical Society Lecture Note Series (No. 345)
Paperback (ISBN-13: 9780521709835)
This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.
? Builds machinery of algebraic geometry upon what students already know well ? Modern approach - focusses on Serre’s GAGA theorem ? Accessible to undergraduates - many examples and exercises throughout the text allow the reader to understand the abstract material at a concrete level
Contents
Foreword; 1. Introduction; 2. Manifolds; 3. Schemes; 4. The complex topology; 5. The analytification of a scheme; 6. The high road to analytification; 7. Coherent sheaves; 8. Projective space - the statements; 9. Projective space - the proofs; 10. The proof of GAGA; Appendix. The proofs concerning analytification; Bibliography; Glossary; Index.
Series: Cambridge Texts in Applied Mathematics (No. 41)
Hardback (ISBN-13: 9780521886802)
A concise account of various classic theories of fluids and solids, this book is for courses in continuum mechanics for graduate students and advanced undergraduates. Thoroughly class-tested in courses at Stanford University and the University of Warwick, it is suitable for both applied mathematicians and engineers. The only prerequisites are an introductory undergraduate knowledge of basic linear algebra and differential equations. Unlike most existing works at this level, this book covers both isothermal and thermal theories. The theories are derived in a unified manner from the fundamental balance laws of continuum mechanics. Intended both for classroom use and for self-study, each chapter contains a wealth of exercises, with fully worked solutions to odd-numbered questions. A complete solutions manual is available to instructors upon request. Short bibliographies appear at the end of each chapter, pointing to material which underpins or expands upon the material discussed.
? Ideal text for advanced undergraduate and graduate students of continuum mechanics; class-tested over 9 years at both Warwick and Stanford Universities ? Contains full solutions to all odd-numbered exercises, with a complete solutions manual available for instructors on request ? Short bibliographies appear at the end of each chapter, pointing to material which underpins and expands upon the material discussed and details further applications ? The text covers both mechanical and thermodynamical theories of fluids and solids ? The book contains numerous examples in the form of fully-worked exercises ? The chapters on tensor algebra and analysis have been carefully written with the beginning student in mind ? The concepts of stress and the Cauchy stress tensor are introduced early and are supported with numerous exercises focused on applications ? The basic balance laws of continuum mechanics and particle mechanics are compared and contrasted so that the beginning student can better a
Contents
Preface; 1. Tensor algebra; 2. Tensor calculus; 3. Continuum mass and force concepts; 4. Kinematics; 5. Balance laws; 6. Isothermal fluid mechanics; 7. Isothermal solid mechanics; 8. Thermal fluid mechanics; 9. Thermal solid mechanics; Answers to selected problems; Bibliography; Index.
Paperback (ISBN-13: 9780521708432)
Given that a college level life science student will take only one additional calculus course after learning the very basics of differentiation and integration, what material should such a course cover? This book answers that question. It is based on a very successful one-semester course taught at Harvard and aims to teach students in the life sciences how to use differential equations to facilitate their research. It requires only a semester's background in calculus. Notions from linear algebra and partial differential equations that are most useful to the life sciences are introduced as and when needed, and in the context of life science applications. In additon, it is designed to teach students how to recognize when differential equations can help focus research. A course taught with this book can replace a standard course in multivariable calculus that is typically to engineers and physicists.
Brought alive by reprints of recent research summary articles from Science and Nature Illustrating the mathematics and demonstrating to students how the mathematics in the text is used by working biologists. Commentary for each reprinted article summarizes the underlying biological issues so neither students nor instructors need prior knowledge, and shows where and how the mathematics is used. Provides students with the mathematics that is commonly used by biologists and life scientistsas opposed to the mathematics of physicists and engineers commonly found in other texts.
Contents
1. Introduction.; 2. Exponential Growth with Appendix on Taylor's Theorem.; 3. Introduction to Differential Equations.; 4. Stability in a One Component System; 5. Systems of First Order Differential Equations; 6. Phase Plane Analysis; 7. Introduction to Vectors; 8. Equilibrium in Two Component, Linear Systems; 9. Stability in Non-Linear Systems; 10. Non-linear Stability Again; 11. Matrix Notation; 12. Remarks about Australian Predators; 13. Introduction to Advection; 14. Diffusion Equations; 15. Two Key Properties of the Advection and Diffusion Equations; 16. The No Trawling Zone; 17. Separation of Variables; 18. The Diffusion Equation and Pattern Formation; 19. Stability Criteria; 20. Summary of Advection and Diffusion; 21. Traveling Waves; 22. Traveling Wave Velocities; 23. Periodic Solutions; 24. Fast and Slow; 25. Estimating Elapsed Time; 26. Switches; 27. Testing for Periodicity; 28. Causes of Chaos. Extra Exercises and Solutions; Index
Series: Cambridge Studies in Advanced Mathematics
Paperback (ISBN-13: 9780521718011)
This is a modern introduction to Kaehlerian geometry and Hodge structure. It starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions. The book is is completely self-contained and can be used by students, while its content gives an up-to-date account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.
? Self-contained with full proofs, so suitable for students ? Only up-to-date treatment of subject ? Contains material never presented before in book form
Contents
Introduction; Part I. Preliminaries: 1. Holomorphic functions of many variables; 2. Complex manifolds; 3. Kahler metrics; 4. Sheaves and cohomology; Part II. The Hodge Decomposition: 5. Harmonic forms and cohomology; 6. The case of Kahler manifolds; 7. Hodge structures and polarisations; 8. Holomorphic de Rham complexes; Part III. Variations of Hodge Structure: 9. Families and deformations; 10. Variations of Hodge structure; Part IV. Cycles and Cycle Classes: 11. Hodge classes; 12. The Abel?Jacobi map; Bibliography; Index.
Review
‘I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.‘ Proceedings of the Edinburgh Mathematical Society
Series: Cambridge Studies in Advanced Mathematics (No. 77)
Paperback (ISBN-13: 9780521718028)
The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard?Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether?Lefschetz theorems, the generic triviality of the Abel?Jacobi maps, and most importantly Nori’s connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.
? Self-contained with full proofs, so suitable for research students ? Only up-to-date treatment of subject ? Contains material never presented before in book form
Contents
Introduction. Part I. The Topology of Algebraic Varieties: 1. The Lefschetz theorem on hyperplane sections; 2. Lefschetz pencils; 3. Monodromy; 4. The Leray spectral sequence; Part II. Variations of Hodge Structure: 5. Transversality and applications; 6. Hodge filtration of hypersurfaces; 7. Normal functions and infinitesimal invariants; 8. Nori’s work; Part III. Algebraic Cycles: 9. Chow groups; 10. Mumford’ theorem and its generalizations; 11. The Bloch conjecture and its generalizations; Bibliography; Index.
Reviews
'All together, the author has maintained her masterly style also throughout this second, much more advanced volume, just as expected. The entire two-volume text is highly instructive, inspiring, reader-friendly and generally outstanding. Without any doubt, these two volumes must be seen as an indispensible standard text on transcendental algebraic geometry for advanced students, teachers, and also researchers in this contemporary field of mathematics. The author provides, simultaneously and in a unique manner, both a complete didactic exposition and an up-to-date presentation of the subject, which is still a rather exceptional feature in the textbook literature. Zentralblatt MATH
'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Burt Totaro, University of Cambridge
‘I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.‘ Proceedings of the Edinburgh Mathematical Society