Algebraic geometry Books

Book SynopsisThe thoroughly revised and updated second edition of this acclaimed text for a second course on linear algebra has more than 1,100 problems and exercises, along with new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices and much more.Trade ReviewReview of the first edition: 'The presentation is straightforward and extremely readable. The authors' enthusiasm pervades the book, and the printing is what we expect from this publisher. This will doubtless be the standard text for years to come.' American ScientistReview of the first edition: 'The reviewer strongly recommends that those working in either pure or applied linear algebra have this book on their desks.' SIAM ReviewReview of the first edition: 'There seems little doubt that the book will become a standard reference for research workers in numerical mathematics.' Computing ReviewsReview of the first edition: 'The authors have done an excellent job of supplying linear algebraists and applied mathematicians with a well-organized comprehensive survey, which can serve both as a text and as a reference.' Linear Algebra and its Applications'The book is well organized, completely readable, and very enlightening. For researchers in matrix analysis, matrix computations, applied linear algebra, or computational science, this second edition is a valuable book.' Jesse L. Barlow, Computing Reviews'With the additional material and exceedingly clear exposition, this book will remain the go-to book for graduate students and researchers alike in the area of linear algebra and matrix theory. I suspect there are few readers who will go through this book and not learn many new things. It is an invaluable reference for anyone working in this area.' Anne Greenbaum, SIAM Review'The new edition is clearly a must-have for anyone seriously interested in matrix analysis.' Nick Higham, Applied Mathematics, Software and Workflow blogTable of Contents1. Eigenvalues, eigenvectors, and similarity; 2. Unitary similarity and unitary equivalence; 3. Canonical forms for similarity, and triangular factorizations; 4. Hermitian matrices, symmetric matrices, and congruences; 5. Norms for vectors and matrices; 6. Location and perturbation of eigenvalues; 7. Positive definite and semi-definite matrices; 8. Positive and nonnegative matrices; Appendix A. Complex numbers; Appendix B. Convex sets and functions; Appendix C. The fundamental theorem of algebra; Appendix D. Continuous dependence of the zeroes of a polynomial on its coefficients; Appendix E. Continuity, compactness, and Weierstrass' theorem; Appendix F. Canonical pairs.

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Book SynopsisDieses Buch will dem Leser eine Einführung in wichtige Techniken und Methoden der heutigen reellen Algebra und Geometrie vermitteln. An Voraussetzungen werden dabei nur Grundkenntnisse der Algebra erwartet, so daß das Buch für Studenten mittlerer Semester geeignet ist.Das erste Kapitel enthält zunächst grundlegende Fakten über angeordnete Körper und ihre reellen Abschlüsse und behandelt dann verschiedene Methoden zur Bestimmung der Anzahl reeller Nullstellen von Polynomen. Das zweite Kapitel befaßt sich mit reellen Stellen und gipfelt in Artins Lösung des 17. Hilbertschen Problems. Kapitel III schließlich ist dem noch jungen Begriff des reellen Spektrums und seinen Anwendungen gewidmet."Neben dem 1987 erschienenen "Géometrie algébrique réelle" von J. Bochnak-M. Coste- M. Roy stellt die vorliegende Monographie das erste Lehrbuch auf diesem Gebiet dar... Damit liegt eine sehr empfehlenswerte Einführung...vor..." (H. Mitsch, Monatshefte für Mathematik 3/111, 1991)Trade Review“More than 30 years after its initial publication, the present textbook is still a very valuable source for results in real algebra. It can serve as a textbook for a university course, but also experts will benefit from the nice account of concepts and results. It’s great that the book is available again, in particular in an English translation for an international audience.” (Tim Netzer, zbMATH 1505.13001, 2023)Table of Contents1 Ordered fields and their real closures.- 2 Convex valuation rings and real places.- 3 The real spectrum.- 4 Recent developments.

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Book SynopsisThe goal of this book is to provide an introduction to algebraic geometry accessible to students. Starting from solutions of polynomial equations, modern tools of the subject soon appear, motivated by how they improve our understanding of geometrical concepts. In many places, analogies and differences with related mathematical areas are explained. The text approaches foundations of algebraic geometry in a complete and self-contained way, also covering the underlying algebra. The last two chapters include a comprehensive treatment of cohomology and discuss some of its applications in algebraic geometry.Trade Review“The prerequisites to read this book are undergraduate analysis, algebra and topology. The proofs of all needed more advanced results of topology or algebra are given. The book is mainly self-contained; some proofs with all the necessary steps are proposed as exercises. … Each chapter ends with a set of exercises.” (Jean-Marc Drézet's, Mathematical Reviews, June, 2022)“The present book is a very solid, and different than usual (or traditional), introduction to algebraic geometry that we can find on the market. … I can honestly recommend this item as a main reference for (advanced) courses devoted to algebraic geometry, and for extended courses devoted to introduction to algebraic geometry – this claim is based on the fact that the book is self-contained on the level of commutative algebra and due to this reason provides a coherent narration.” (Piotr Pokora, zbMATH 1471.14003, 2021)Table of ContentsPreface.- Introduction.- Beginning concepts.- Schemes.- Properties of schemes.- Sheaves of modules.- Introduction to Cohomology.- Cohomology in algebraic geometry.- Exercises.

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Book SynopsisThis text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text.Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.Trade Review“The textbook is a concise and well organized treatment of characteristic classes on principal bundles. It is characterized by a right balance between rigor and simplicity. It should be in every mathematician's arsenal and take its place in any mathematical library.” (Nabil L. Youssef, zbMATH 1383.53001, 2018)Table of ContentsPreface.- Chapter 1. Curvature and Vector Fields.- 1. Riemannian Manifolds.- 2. Curves.- 3. Surfaces in Space.- 4. Directional Derivative in Euclidean Space.- 5. The Shape Operator.- 6. Affine Connections.- 7. Vector Bundles.- 8. Gauss's Theorema Egregium.- 9. Generalizations to Hypersurfaces in Rn+1.- Chapter 2. Curvature and Differential Forms.- 10. Connections on a Vector Bundle.- 11. Connection, Curvature, and Torsion Forms.- 12. The Theorema Egregium Using Forms.- Chapter 3. Geodesics.- 13. More on Affine Connections.- 14. Geodesics.- 15. Exponential Maps.- 16. Distance and Volume.- 17. The Gauss-Bonnet Theorem.- Chapter 4. Tools from Algebra and Topology.- 18. The Tensor Product and the Dual Module.- 19. The Exterior Power.- 20. Operations on Vector Bundles.- 21. Vector-Valued Forms.- Chapter 5. Vector Bundles and Characteristic Classes.- 22. Connections and Curvature Again.- 23. Characteristic Classes.- 24. Pontrjagin Classes.- 25. The Euler Class and Chern Classes.- 26. Some Applications of Characteristic Classes.- Chapter 6. Principal Bundles and Characteristic Classes.- 27. Principal Bundles.- 28. Connections on a Principal Bundle.- 29. Horizontal Distributions on a Frame Bundle.- 30. Curvature on a Principal Bundle.- 31. Covariant Derivative on a Principal Bundle.- 32. Character Classes of Principal Bundles.- A. Manifolds.- B. Invariant Polynomials.- Hints and Solutions to Selected End-of-Section Problems.- List of Notations.- References.- Index.

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Book SynopsisThis textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises). In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading. A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.Trade Review“This book contains a collection of exercises in Galois theory. … The book provides the readers with a solid exercise-based introduction to classical Galois theory; it will be useful for self-study or for supporting a lecture course.” (Franz Lemmermeyer, zbMATH 1396.12001, 2018)Table of Contents1 Solving algebraic equations.- 2 Field extensions.- 3 Polynomials and irreducibility.- 4 Algebraic extensions.- 5 Splitting fields.- 6 Automorphism groups of fields.- 7 Normal extensions.- 8 Separable extensions.- 9 Galois extensions.- 10 Cyclotomic extensions.- 11 Galois modules.- 12 Solvable groups.- 13 Solvability of equations.- 14 Geometric constructions.- 15 Computing Galois groups.- 16 Supplementary problems.- 17 Proofs of the theorems.- 18 Hints and answers.- 19 Examples and selected solutions.- Appendix: Groups, rings and fields.- References.- List of notations.- Index.

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Book SynopsisPresents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized.Table of Contents A crash course in commutative algebra Affine varieties Projective varieties Regular and rational maps of quasi-projective varieties Products The blow-up of an ideal Finite maps of quasi-projective varieties Dimension of quasi-projective algebraic sets Zariski's main theorem Nonsingularity Sheaves Applications to regular and rational maps Divisors Differential forms and the canonical divisor Schemes The degree of a projective variety Cohomology Curves An introduction to intersection theory Surfaces Ramification and etale maps Bertini's theorem and general fibers of maps Bibliography Index.

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Book SynopsisThis 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;Trade ReviewFrom the reviews:“The textbook under review provides a modern introduction to the theory of modular forms, with the aim to explain the modularity theorem to beginning graduate students and advanced undergraduates. … Written in a very comprehensible, detailed, lucid and instructive manner, this unique textbook is widely self-contained and perfectly suitable for self-study by beginners. … an excellent guide to the relevant research literature … . experts and teachers will get a lot of methodological inspiration from the authors’ approach, and many useful ideas for efficient teaching.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, June, 2013)"It has always been difficult to start learning about modular forms. … we were still lacking a textbook that could be honestly described as both comprehensive and accessible. Diamond and Shurman’s First Course is a largely successful attempt to provide just such a book. … A First Course in Modular Forms is a success. … a course taught from this text would be a very good way to lead students into the area. … I expect that Diamond and Shurman’s book would serve very well." (Fernando Q. Gouvêa, MathDL, February, 2007)"An essentially self-contained treatment that readers will find valuable both as a reference and a pedagogical text. ... The authors of FCMF are to be commended for producing a valuable addition to the literature which belongs on the shelf of all scholars with an interest in modular forms, modular curves and their arithmetic applications." (Henri Darmon, Mathematical Reviews, Issue 2006 f)"The aim of this book is to introduce the reader to the modularity theorem. … This book can be recommended to everyone wishing to learn about modular forms and their connections to number theory." (J. Mahnkopf, Monatshefte für Mathematik, Vol. 146 (4), 2006)"The … goal of Diamond (Brandeis Univ.) and Shurman (Reed College) is … to state the modularity conjecture in some of its many forms. … readers wishing eventually to read Wiles could hardly find a better place to start than this. … Summing Up: Highly recommended. General readers; upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. 43 (1), September, 2005)"The textbook under review provides a modern introduction to the theory of modular forms … . This ambitious program … is carried out in as down-to-earth a way as possible. … this is the first comprehensive introduction to the recent modularity theorem … . Written in a very comprehensible, detailed, lucid and instructive manner, this unique textbook is widely self-contained and perfectly suitable for self-study by beginners. Moreover, this book is an excellent guide to the relevant research literature … ." (Werner Kleinert, Zentralblatt MATH, Vol. 1062 (13), 2005)"While there are many books on modular forms and elliptic curves, and some of them discuss the Eicheler-Shimura theory, most that describe it do not go deeply into the proofs. … The book of Diamond and Shurman addresses this need. … it is clearly directed to the serious student and it will unquestionably be a useful book even to experts. … this is a very unique and valuable book, and one that I would recommend to anyone wishing to learn about modular forms … ." (Daniel Bump, SIAM Review, Vol. 47 (4), 2005)"This introduction to modular forms is aimed at students with only a basic knowledge of complex function theory. … A useful and up-to-date exposition of topics scattered throughout the literature, aided by exercises with answers." (Mathematika, Vol. 52, 2005)Table of ContentsModular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves as Algebraic Curves.- The Eichler-Shimura Relation and L-functions.- Galois Representations.

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Book SynopsisRapid, concise, self-contained introduction assumes only familiarity with elementary algebra. Subjects include algebraic varieties; products, projections, and correspondences; normal varieties; differential forms; theory of simple points; algebraic groups; more. 1958 edition.

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Book SynopsisSeventeen articles from the most outstanding contemporary topics in algebraic geometry. Includes a beautiful exposition of the new simple approach to the resolution of singularities, a detailed essay on the A,D,E classification, a discussion of the new special Lagrangian approach to mirror symmetry, and two surveys of Gromow-Witten invariants.Table of Contents1. On the Betti number of birationally isomorphic projective varieties with trivial canonical bundles V. V. Batyrev; 2. A Calabi–Yau threefold with non-abelian fundamental group A. Beauville; 3. Introduction to the theory of Gromov–Witten K. Behrend; 4. Kähler-hyperbolicity and variations of Hodge structures Philippe Eyssidieux; 5. Algorithms for computing intersections numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians C. Faber; 6. On some tensor representations of the Cremona group of the projective plane M. Gizatullin; 7. Hilbert schemes and simple singularities Y. Ito; 8. Bounds for Seshadri constants O. Küchle; 9. Degenerate double covers of the projective plane M. Manetti; 10. The geometry underlying mirror symmetry D. R. Morrison; 11. Duality of polarized K3 surfaces S. Mukai; 12. Hilbert schemes and simple singularities I. Nakamura; 13. On symplectic invariants of algebraic varieties coming from crepant contractions R. Paoletti; 14. Bogomolov-Pantev resolution - an expository account K. H. Paranjape; 15. Symplectic Gromov–Witten invariants B. Siebert; 16. Mordell–Weil lattices for higher genus fibration over a curve T. Shioda; 17. Bounds for Seshadri constants A. Steffens; 18. A generic Torelli theorem for the quintic threefold C. Voisin; 19. Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau threefolds P. M. H. Wilson.

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Book SynopsisConsiders the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. This book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension.Trade Review"Zannier's book is well written and a pleasure to read... [T]he author always makes an effort to point out key ideas and key steps, so a reader who wants to read and understand the complete proofs in this technically demanding field will find this monograph to be an extremely helpful entree into the subject... [T]he reviewer highly recommends Zannier's book as an excellent survey of and introduction to the important and hot topic of unlikely intersections in arithmetic geometry."--Joseph H. Silverman, Bulletin of the AMS "This book is indeed a great source of knowledge and inspiration for everybody interested in the unlikely intersection problems. The author must be commended for doing this job, and doing it so well."--Yuri Bilu, Mathematical Reviews ClippingsTable of Contents*FrontMatter, pg. i*Contents, pg. v*Preface, pg. ix*Notation and Conventions, pg. xi*Introduction: An Overview of Some Problems of Unlikely Intersections, pg. 1*Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture, pg. 15*Chapter 2: An Arithmetical Analogue, pg. 43*Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser, pg. 62*Chapter 4: About the Andre-Oort Conjecture, pg. 96*Appendix A: Distribution of Rational Points on Subanalytic Surfaces, pg. 128*Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions, pg. 136*Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof, pg. 138*Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach, pg. 140*Appendix E: A Transcendence Measure for a Quotient of Periods, pg. 143*Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach, pg. 145*Appendix G: Mixed Problems: Another Approach, pg. 147*Bibliography, pg. 149*Index, pg. 159

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Book SynopsisThis volume uses a unified approach to representation theory and automorphic forms.Table of ContentsIntroduction.- Ramakrishnan, D.: Irreducibility and Cuspidality.-Ikeda, T.: On Liftings of Holomorphic Modular Forms.-Kobayashi, T.: Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs.-Miller, S., Schmid, W.: The Rankin--Selberg Method for Automorphic Distributions.- Shahidi, F.: Langlands Functoriality Conjecture and Number Theory.- Yoshikawa, K.: Discriminant of certain K3 surfaces.- References.- Index.

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Book SynopsisPoint-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the AndréOort and ZilberPink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate studTable of Contents1. Introduction; Part I. Point-Counting and Diophantine Applications: 2. Point-counting; 3. Multiplicative Manin–Mumford; 4. Powers of the Modular Curve as Shimura Varieties; 5. Modular André–Oort; 6. Point-Counting and the André–Oort Conjecture; Part II. O-Minimality and Point-Counting: 7. Model theory and definable sets; 8. O-minimal structures; 9. Parameterization and point-counting; 10. Better bounds; 11. Point-counting and Galois orbit bounds; 12. Complex analysis in O-minimal structures; Part III. Ax–Schanuel Properties: 13. Schanuel's conjecture and Ax–Schanuel; 14. A formal setting; 15. Modular Ax–Schanuel; 16. Ax–Schanuel for Shimura varieties; 17. Quasi-periods of elliptic curves; Part IV. The Zilber–Pink Conjecture: 18. Sources; 19. Formulations; 20. Some results; 21. Curves in a power of the modular curve; 22. Conditional modular Zilber–Pink; 23. O-minimal uniformity; 24. Uniform Zilber–Pink; References; List of notation; Index.

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Book SynopsisA First Course in Computational Algebraic Geometry is designed for young students with some background in algebra who wish to perform their first experiments in computational geometry. Originating from a course taught at the African Institute for Mathematical Sciences, the book gives a compact presentation of the basic theory, with particular emphasis on explicit computational examples using the freely available computer algebra system, Singular. Readers will quickly gain the confidence to begin performing their own experiments.Trade Review'Decker and Pfister exposit the very rudiments of algebraic geometry in tandem with the workings of the program Singular. The student who masters this content could proceed in many directions … Recommended. Upper-division undergraduates through researchers/faculty.' D. V. Feldman, ChoiceTable of ContentsPreface; Prologue: general remarks on computer algebra systems; 1. The geometry-algebra dictionary; 2. Computing; 3. Sudoku; 4. A problem in group theory solved by computer algebra; Bibliography; Index.

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Book SynopsisPresents selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held in April 2018. Topics include an introduction to a new class of quiver algebras on surfaces called ‘geodesic ghor algebras’, and a presentation of Feynman categories from a representation-theoretic viewpoint.Table of Contents K. Baur and C. Beil, Examples of geodesic ghor algebras on hyperbolic surfaces R. M. Kaufmann, Feynman categories and representation theory M. Kleiner, Preprojective roots and graph monoids of Coxeter groups A. Neeman, Approximable triangulated categories S. Posur, Methods of constructive category theory C. E. Parra and M. Saorin, The HRS tilting process and Grothendieck hearts of t-structures

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Book SynopsisIllustrates the fruitful interaction between abstract theory and explicit computations, covering a large range of topics, including Belyi maps, Galois representations attached to elliptic curves, reconstruction of curves from their Jacobians, isogeny graphs of abelian varieties, hypergeometric equations, and Drinfeld modules.

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Book SynopsisThe most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.Trade Review""This is a wonderful introduction...You may end up amazed and incredulous." - Leon M. Lederman, Nobel Laureate“The mathematics in this book is a delight: surprising, insightful, and comprehensive… the result is by turns rigorous, entertaining, and eye-poppingly speculative.”-New Scientist “…a work that, although aimed at a general audience and presupposing no knowledge of mathematics beyond the high school precalculus level, succeeds in entertaining all audiences…Educators, as well as the mathematically curious, are encouraged to pick up this volume. The discussions of Fibonacci numbers in nature, art, architecture, and music are very thorough…highly recommended.” -Choice“The authors have breathed life into what could be considered a fairly dry subject by demonstrating how commonplace items make use of the Fibonacci numbers…there is a great deal of math involved but taken step at a time, it is not that difficult to understand and this understanding leads to a an even greater appreciation of everything from a flower garden to classical music. Overall, this is an interesting if challenging read for the layperson and a gold mine for the mathematically inclined.”-Monsters and Critics “…the authors have presented a compelling and well-developed book, and one that might well make converts out of some hard-core math phobics…an elegant book that enhances their argument that mathematics is ‘the queen of sciences'.”-Education Update “…delightful…accessible to anyone who enjoys or enjoyed high school mathematics. Mathematics teachers from middle school through college will find this book fun to read and useful in the classroom. The authors consider more properties, relationships, and applications of the Fibonacci numbers than most other sources do…I enjoyed reading this book…a valuable addition to the mathematical literature.”-Mathematics Teacher

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Book SynopsisThis book provides a gentle introduction to the foundations of Algebraic Geometry, starting from computational topics (ideals and homogeneous ideals, zero loci of ideals) up to increasingly intrinsic and abstract arguments, such as 'Algebraic Varieties', whose natural continuation is a more advanced course on the theory of schemes, vector bundles, and sheaf-cohomology.Valuable to students studying Algebraic Geometry and Geometry, this title contains around 60 exercises (with solutions) to help students thoroughly understand the theories introduced in the book. Proofs of the results are carried out in full detail. Many examples are discussed in order to reinforce the understanding of both the theoretical elements and their consequences, as well as the possible applications of the material.

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Book SynopsisThis book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.Trade Review“This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc.” (Piotr Pokora, zbMATH 1445.13001, 2020)Table of Contents- Part I Associated Primes of Powers of Ideals - Associated Primes of Powers of Ideals. - Associated Primes of Powers of Squarefree Monomial Ideals. - Final Comments and Further Reading. - Part II Regularity of Powers of Ideals. - Regularity of Powers of Ideals and the Combinatorial Framework. - Problems, Questions, and Inductive Techniques. - Examples of the Inductive Techniques. - Final Comments and Further Reading. - Part III The Containment Problem. - The Containment Problem: Background. - The Containment Problem. - The Waldschmidt Constant of Squarefree Monomial Ideals. - Symbolic Defect. - Final Comments and Further Reading. - Part IV Unexpected Hypersurfaces. - Unexpected Hypersurfaces. - Final Comments and Further Reading.

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Book SynopsisLa géométrie rigide est devenue, au fil des ans, un outil indispensable dans un grand nombre de questions en géométrie arithmétique. Depuis ses premières fondations, posées par J. Tate en 1961, la théorie s'est développée dans des directions variées. Ce livre est le premier volume d'un traité qui expose un développement systématique de la géométrie rigide suivant l'approche de M. Raynaud, basée sur les schémas formels à éclatements admissibles près. Ce volume est consacré à la construction des espaces rigides dans une situation relative et à l'étude de leurs propriétés géométriques. L'accent est particulièrement mis sur l'étude de la topologie admissible d'un espace rigide cohérent, analogue de la topologie de Zariski d'un schéma. Parmi les sujets traités figurent l'étude des faisceaux cohérents et de leur cohomologie, le théorème de platification par éclatements admissibles qui généralise au cadre formel-rigide un théorème de Raynaud-Gruson dans le cadre algébrique, et le théorème de comparaison du type GAGA pour les faisceaux cohérents. Ce volume contient aussi de larges rappels et compléments de la théorie des schémas formels de Grothendieck. Ce traité est destiné tout autant aux étudiants ayant une bonne connaissance de la géométrie algébrique et souhaitant apprendre la géométrie rigide qu'aux experts en géométrie algébrique et en théorie des nombres comme source de références. Table of ContentsPréface par Michel Raynaud.- Avant-propos.- Introduction.- Chapitre 1. Préliminaires.- Chapitre 2. Géométrie formelle.- Chapitre 3. Éclatements admissibles.- Chapitre 4. Géométrie rigide.- Chapitre 5. Platitude.- Chapitre 6. Invariants différentiels. Morphismes lisses.- Chapitre 7. Espaces rigides quasi-séparés.- Bibliographie.- Index.

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Book SynopsisThis book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.Table of Contents1. Notes on Weyl Algebras and D-modules.- 2. Inverse Systems of Local Rings.- 3. Lectures on the Representation Type of a Projective Variety.- 4. Simplicial Toric Varieties which are set-theoretic Complete Intersections.

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Book SynopsisThe first survey of its kind, written by internationally known, outstanding experts who developed substantial parts of the field. The book contains an introduction written by Remmert, describing the history of the subject, and is very useful to graduate students and researchers in complex analysis, algebraic geometry and differential geometry.Table of ContentsI. Local Theory of Complex Spaces.- II. Differential Calculus, Holomorphic Maps and Linear Structures on Complex Spaces.- III. Cohomology.- IV. Seminormal Complex Spaces.- V. Pseudoconvexity, the Levi Problem and Vanishing Theorems.- VI. Theory of q-Convexity and q-Concavity.- VII. Modifications.- VIII. Cycle Spaces.- IX. Extension of Analytic Objects.- Author Index.

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Book SynopsisProvides an easy to understand mathematical tool set for professionals an students in electromagnetic study Non-axiomatic, non-challenging, less formal tutorial approach on the subject Includes appendices with reference material that includes a helpful glossary of terms .Trade Review"This book will benefit scientists and engineers who use electromagnetic theory in the course of their work.” (Zentralblatt MATH, 1 May 2013)Table of ContentsPreface xi Reading Guide xv 1. Introduction 1 2. A Quick Tour of Geometric Algebra 7 2.1 The Basic Rules of a Geometric Algebra 16 2.2 3D Geometric Algebra 17 2.3 Developing the Rules 19 2.3.1 General Rules 20 2.3.2 3D 21 2.3.3 The Geometric Interpretation of Inner and Outer Products 22 2.4 Comparison with Traditional 3D Tools 24 2.5 New Possibilities 24 2.6 Exercises 26 3. Applying the Abstraction 27 3.1 Space and Time 27 3.2 Electromagnetics 28 3.2.1 The Electromagnetic Field 28 3.2.2 Electric and Magnetic Dipoles 30 3.3 The Vector Derivative 32 3.4 The Integral Equations 34 3.5 The Role of the Dual 36 3.6 Exercises 37 4. Generalization 39 4.1 Homogeneous and Inhomogeneous Multivectors 40 4.2 Blades 40 4.3 Reversal 42 4.4 Maximum Grade 43 4.5 Inner and Outer Products Involving a Multivector 44 4.6 Inner and Outer Products between Higher Grades 48 4.7 Summary So Far 50 4.8 Exercises 51 5. (3+1)D Electromagnetics 55 5.1 The Lorentz Force 55 5.2 Maxwell’s Equations in Free Space 56 5.3 Simplifi ed Equations 59 5.4 The Connection between the Electric and Magnetic Fields 60 5.5 Plane Electromagnetic Waves 64 5.6 Charge Conservation 68 5.7 Multivector Potential 69 5.7.1 The Potential of a Moving Charge 70 5.8 Energy and Momentum 76 5.9 Maxwell’s Equations in Polarizable Media 78 5.9.1 Boundary Conditions at an Interface 84 5.10 Exercises 88 6. Review of (3+1)D 91 7. Introducing Spacetime 97 7.1 Background and Key Concepts 98 7.2 Time as a Vector 102 7.3 The Spacetime Basis Elements 104 7.3.1 Spatial and Temporal Vectors 106 7.4 Basic Operations 109 7.5 Velocity 111 7.6 Different Basis Vectors and Frames 112 7.7 Events and Histories 115 7.7.1 Events 115 7.7.2 Histories 115 7.7.3 Straight-Line Histories and Their Time Vectors 116 7.7.4 Arbitrary Histories 119 7.8 The Spacetime Form of ∇ 121 7.9 Working with Vector Differentiation 123 7.10 Working without Basis Vectors 124 7.11 Classifi cation of Spacetime Vectors and Bivectors 126 7.12 Exercises 127 8. Relating Spacetime to (3+1)D 129 8.1 The Correspondence between the Elements 129 8.1.1 The Even Elements of Spacetime 130 8.1.2 The Odd Elements of Spacetime 131 8.1.3 From (3+1)D to Spacetime 132 8.2 Translations in General 133 8.2.1 Vectors 133 8.2.2 Bivectors 135 8.2.3 Trivectors 136 8.3 Introduction to Spacetime Splits 137 8.4 Some Important Spacetime Splits 140 8.4.1 Time 140 8.4.2 Velocity 141 8.4.3 Vector Derivatives 142 8.4.4 Vector Derivatives of General Multivectors 144 8.5 What Next? 144 8.6 Exercises 145 9. Change of Basis Vectors 147 9.1 Linear Transformations 147 9.2 Relationship to Geometric Algebras 149 9.3 Implementing Spatial Rotations and the Lorentz Transformation 150 9.4 Lorentz Transformation of the Basis Vectors 153 9.5 Lorentz Transformation of the Basis Bivectors 155 9.6 Transformation of the Unit Scalar and Pseudoscalar 156 9.7 Reverse Lorentz Transformation 156 9.8 The Lorentz Transformation with Vectors in Component Form 158 9.8.1 Transformation of a Vector versus a Transformation of Basis 158 9.8.2 Transformation of Basis for Any Given Vector 162 9.9 Dilations 165 9.10 Exercises 166 10. Further Spacetime Concepts 169 10.1 Review of Frames and Time Vectors 169 10.2 Frames in General 171 10.3 Maps and Grids 173 10.4 Proper Time 175 10.5 Proper Velocity 176 10.6 Relative Vectors and Paravectors 178 10.6.1 Geometric Interpretation of the Spacetime Split 179 10.6.2 Relative Basis Vectors 183 10.6.3 Evaluating Relative Vectors 185 10.6.4 Relative Vectors Involving Parameters 188 10.6.5 Transforming Relative Vectors and Paravectors to a Different Frame 190 10.7 Frame-Dependent versus Frame-Independent Scalars 192 10.8 Change of Basis for Any Object in Component Form 194 10.9 Velocity as Seen in Different Frames 196 10.10 Frame-Free Form of the Lorentz Transformation 200 10.11 Exercises 202 11. Application of the Spacetime Geometric Algebra to Basic Electromagnetics 203 11.1 The Vector Potential and Some Spacetime Splits 204 11.2 Maxwell’s Equations in Spacetime Form 208 11.2.1 Maxwell’s Free Space or Microscopic Equation 208 11.2.2 Maxwell’s Equations in Polarizable Media 210 11.3 Charge Conservation and the Wave Equation 212 11.4 Plane Electromagnetic Waves 213 11.5 Transformation of the Electromagnetic Field 217 11.5.1 A General Spacetime Split for F 217 11.5.2 Maxwell’s Equation in a Different Frame 219 11.5.3 Transformation of F by Replacement of Basis Elements 221 11.5.4 The Electromagnetic Field of a Plane Wave Under a Change of Frame 223 11.6 Lorentz Force 224 11.7 The Spacetime Approach to Electrodynamics 227 11.8 The Electromagnetic Field of a Moving Point Charge 232 11.8.1 General Spacetime Form of a Charge’s Electromagnetic Potential 232 11.8.2 Electromagnetic Potential of a Point Charge in Uniform Motion 234 11.8.3 Electromagnetic Field of a Point Charge in Uniform Motion 237 11.9 Exercises 240 12. The Electromagnetic Field of a Point Charge Undergoing Acceleration 243 12.1 Working with Null Vectors 243 12.2 Finding F for a Moving Point Charge 248 12.3 Frad in the Charge’s Rest Frame 252 12.4 Frad in the Observer’s Rest Frame 254 12.5 Exercises 258 13. Conclusion 259 14. Appendices 265 14.1 Glossary 265 14.2 Axial versus True Vectors 273 14.3 Complex Numbers and the 2D Geometric Algebra 274 14.4 The Structure of Vector Spaces and Geometric Algebras 275 14.4.1 A Vector Space 275 14.4.2 A Geometric Algebra 275 14.5 Quaternions Compared 281 14.6 Evaluation of an Integral in Equation (5.14) 283 14.7 Formal Derivation of the Spacetime Vector Derivative 284 References 287 Further Reading 291 Index 293 The IEEE Press Series on Electromagnetic Wave Theory

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Book SynopsisA comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric.Table of ContentsFoundational Material. Complex Algebraic Varieties. Riemann Surfaces and Algebraic Curves. Further Techniques. Surfaces. Residues. The Quadric Line Complex. Index.

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Book SynopsisTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. ix*Terminology and Conventions, pg. xiii*Chapter I. Etale Morphisms, pg. 1*Chapter II. Sheaf Theory, pg. 46*Chapter III. Cohomology, pg. 82*Chapter IV. The Brauer Group, pg. 136*Chapter V. The Cohomology of Curves and Surfaces, pg. 155*Chapter VI. The Fundamental Theorems, pg. 220*Appendix A. Limits, pg. 304*Appendix B. Spectral Sequences, pg. 307*Appendix C. Hypercohomology, pg. 310*Bibliography, pg. 313*Index, pg. 321

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Book SynopsisThis book, a translation of the German volume n-Ecke, presents an elegant geometric theory which, starting from quite elementary geometrical observations, exhibits an interesting connection between geometry and fundamental ideas of modern algebra in a form that is easily accessible to the student who lacks a sophisticated background in mathematics. It stimulates geometrical thought by applying the tools of linear algebra and the algebra of polynomials to a concrete geometrical situation to reveal some rather surprising insights into the geometry of n-gons. The twelve chapters treat n-gons, classes of n-gons, and mapping of the set of n-gons into itself. Exercises are included throughout, and two appendixes, by Henner Kinder and Eckart Schmidt, provide background material on lattices and cyclotomic polynomials.(Mathematical Expositions No. 18)

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Book SynopsisIntroduces the world of advanced mathematics using algebraic structures as a unifying theme. Having no prerequisites beyond precalculus and an interest in abstract reasoning, the book is suitable for students who are looking for an entry into discrete mathematics, induction and recursion, groups and symmetry, and plane geometry.Table of Contents Logic and proofs An introduction to sets The integers Mappings and relations Induction and recursion Binary operations Groups Divisibility and congruences Primes Multiplicative inverses of residue classes Linear transformations Isomorphism The symmetric group Examples of finite groups Cosets Homomorphisms Group actions Euclidean geometry Euler's formula Index

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Book SynopsisPresents survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc, in which cyclic homology plays the role of a unifying theme.Table of Contents A. Baldare, M. Benameur, and V. Nistor, Chern-Connes-Karoubi character isomorphisms and algebras of symbols of pseudodifferential operators J. Block, N. Higson, and J. Sanchez Jr., On Perrot's index cocycles P. Carrillo Rouse, The Chern-Baum Connes assembly map for Lie groupoids A. Connes and C. Consani, Hochschild homology, trace map and $\zeta$-cycles A. Connes and C. Consani, Cyclic theory and the pericyclic category J. Cuntz, The image of Bott peridocity in cyclic homology B. I. Dundas, Applications of topological cyclic homology to algebraic $K$-theory D. Gepner, Algebraic $K$-theory and generalized stable homotopy theory A. Gorokhovsky and E. van Erp, Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrose L. Hesselholt, Topological cyclic homology and the Fargues-Fontaine curve M. Khalkhali and I. Shapiro, Hopf cyclic cohomology and beyond M. Lorentz, The Hochschild cohomology of uniform Roe algebras E. McDonald, F. Sukochev, and X. Xiong, Quantum differntiability-The analytical perspective R. Meyer and D. Mukherjee, Local cyclic homology for nonarchimedean Banach algebras H. Moscovici, On the van Est analogy in Hopf cyclic cohomology P. Piazza and X. Tang, Primary and secondary invariants of Dirac operators on $G$-proper manifolds M. J. Pflaum, Localization in Hochschild homology R. Ponge, Cyclic homology and group actions E. Prodan, Cyclic cocycles and quantized pairings in materials science M. Puschnigg, Periodic cyclic homology of crossed products A. Savin and E. Schrohe, Trace expansions and equivariant traces on an algebra of Fourier integral operators on $\mathbb{R}^n$ Y. Song and X. Tang, Carton motion group and orbital integrals B. Tsygan, On noncommutative crystalline cohomology T. D. H. Van Nuland and W. D. van Suijlekom, Cyclic cocycles and one-loop corrections in the spectral action J. Wang, Z. Xie, and G. Yu, $\ell^1$-higher index, $\ell^1$-higher rho invariant and cyclic cohomology.

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Book SynopsisProgress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's λ-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his λ-invariant. The principal goal of this book, now in its second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and its applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds.Trade ReviewThis is an excellent introduction to the Rokhlin and Casson invariants for homology 3-spheres [...], and in particular also to the necessary background material from the theory of 3- and 4-manifolds [...], so the book may serve also as a reasonable short and efficient introduction to some important parts of low-dimensional topology. It grew out of a course for second year graduate students and concentrates 19 lectures on less than 200 pages, including also a glossary on back-ground material from algebraic topology, a collection of exercises, open problems and comments on recent developments [...] To conclude, the author has succeeded in presenting a lot of material in a clear and efficient way, and the book is interesting and stimulating to read. Birge Zimmermann-Huisgen, Zentralblatt MATHTable of ContentsPreface Introduction Glossary 1 Heegaard splittings 1.1 Introduction 1.2 Existence of Heegaard splittings 1.3 Stable equivalence of Heegaard splittings 1.4 The mapping class group 1.5 Manifolds of Heegaard genus ≤ 1 1.6 Seifert manifolds 1.7 Heegaard diagrams 2 Dehn surgery 2.1 Knots and links in 3-manifolds 2.2 Surgery on links in S3 2.3 Surgery description of lens spaces and Seifert manifolds 2.4 Surgery and 4-manifolds 3 Kirby calculus 3.1 The linking number 3.2 Kirby moves 3.3 The linking matrix 3.4 Reversing orientation 4 Even surgeries 5 Review of 4-manifolds 5.1 Definition of the intersection form 5.2 The unimodular integral forms 5.3 Four-manifolds and intersection forms 6 Four-manifolds with boundary 6.1 The intersection form 6.2 Homology spheres via surgery on knots 6.3 Seifert homology spheres 6.4 The Rohlin invariant 7 Invariants of knots and links 7.1 Seifert surfaces 7.2 Seifert matrices 7.3 The Alexander polynomial 7.4 Other invariants from Seifert surfaces 7.5 Knots in homology spheres 7.6 Boundary links and the Alexander polynomial 8 Fibered knots 8.1 The definition of a fibered knot 8.2 The monodromy 8.3 More about torus knots 8.4 Joins 8.5 The monodromy of torus knots 8.6 Open book decompositions 9 The Arf-invariant 9.1 The Arf-invariant of a quadratic form 9.2 The Arf-invariant of a knot 10 Rohlin’s theorem 10.1 Characteristic surfaces 10.2 The definition of ˜q 10.3 Representing homology classes by surfaces 11 The Rohlin invariant 11.1 Definition of the Rohlin invariant 11.2 The Rohlin invariant of Seifert spheres 11.3 A surgery formula for the Rohlin invariant 11.4 The homology cobordism group 12 The Casson invariant 13 The group SU(2) 14 Representation spaces 14.1 The topology of representation spaces 14.2 Irreducible representations 14.3 Representations of free groups 14.4 Representations of surface groups 14.5 Representations for Seifert homology spheres 15 The local properties of representation spaces 16 Casson’s invariant for Heegaard splittings 16.1 The intersection product 16.2 The orientations 16.3 Independence of Heegaard splitting 17 Casson’s invariant for knots 17.1 Preferred Heegaard splittings 17.2 The Casson invariant for knots 17.3 The difference cycle 17.4 The Casson invariant for boundary links 17.5 The Casson invariant of a trefoil 18 An application of the Casson invariant 18.1 Triangulating 4-manifolds 18.2 Higher-dimensional manifolds 19 The Casson invariant of Seifert manifolds 19.1 The space R(p; q; r) 19.2 Calculation of the Casson invariant Conclusion Bibliography Index

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Book SynopsisThis book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

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Book SynopsisWhile Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer’s fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi’s important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kähler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi’s mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi’s visionary contributions will certainly continue to shape the course of this subject far into the future.Trade Review“In my case, I spent several happy hours learning about affine differential geometry, something that would certainly never have happened if I had not picked up this volume. … The collected works of Eugenio Calabi are worthy of a place on the bookshelf of any person with a serious interest in differential geometry.” (Joel Fine, EMS Magazine, May 11, 2023)Table of ContentsPreface.- J.-P. Bourguignon, Eugenio Calabi’s Short Biography.- Bibliographic List of Works.- S.-T. Yau, An Essay on Eugenio Calabi.- Part I: Commentaries on Calabi’s Life and Work: B. Lawson, Reflections on the Early Work of Eugenio Calabi.- M. Berger, Encounter with a Geometer: Eugenio Calabi.- J.-P. Bourguignon, Eugenio Calabi and Kähler Metrics.- C. LeBrun, Eugenio Calabi and the Curvature of Kähler Manifolds.- X. Chen, S. Donaldson, Calabi’s Work on Affine Differential Geometry and Results of Bernstein Type.- Part II: Collected Works: E. Calabi ,Ar. Dvoretzky, Convergence- and Sum-Factors for Series of Complex Numbers (1951).- E. Calabi, D. C. Spencer, Completely Integrable Almost Complex Manifolds (1951).- E. Calabi, Metric Riemann Surfaces (1953).- E. Calabi, M. Rosenlicht, Complex Analytic Manifolds Without Countable Base (1953).- E. Calabi, B. Eckmann, A Class of Compact, Complex Manifolds Which Are Not Algebraic (1953).- E. Calabi, Isometric Imbedding of Complex Manifolds (1953).- E. Calabi, The Space of Kähler Metrics (1954).- E. Calabi, The Variation of Kähler Metrics I. The Structure of the Space (1954).- E. Calabi, The Variation of Kähler Metrics II. A Minimum Problem (1954).- E. Calabi, On Kähler Manifolds With Vanishing Canonical Class (1957).- E. Calabi, Construction and Properties of Some 6-Dimensional Almost Complex Manifolds (1958).- E. Calabi, Improper Affine Hyperspheres of Convex Type and a Generalization of a Theorem by K. Jörgens (1958).- E. Calabi, An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1958).- E. Calabi, Errata: An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1959).- E. Calabi, E. Vesentini, Sur les variétés complexes compactes localement symétriques (1959).- E. Calabi, E. Vesentini, On Compact, Locally Symmetric Kähler Manifolds (1960).- E. Calabi, On Compact, Riemannian Manifolds with Constant Curvature I. (1961).- E. Calabi, L. Markus Relativistic Space Forms (1962).- E. Calabi, Linear Systems of Real Quadratic Forms (1964).- E. Calabi, Quasi-Surjective Mappings and a Generalization of Morse Theory (1966).- E. Calabi, Minimal Immersions of Surfaces in Euclidean Spheres (1967).- E. Calabi, On Ricci Curvature and Geodesics (1967).- E. Calabi, On Differentiable Actions of Compact Lie Groups on Compact Manifolds (1968).- E. Calabi, An Intrinsic Characterization of Harmonic One-Forms (1969).- E. Calabi, On the Group of Automorphisms of a Symplectic Manifold (1970).- E. Calabi, P. Hartman, On the Smoothness of Isometries (1970).- E. Calabi, Examples of Bernstein Problems for Some Nonlinear Equations (1970).- E. Calabi, Über singuläre symplektische Strukturen (1971).- E. Calabi, Complete Affine Hyperspheres I (1972).- E. Calabi, A Construction of Nonhomogeneous Einstein Metrics (1975).- E. Calabi, H. S. Wilf, On the Sequential and Random Selection of Subspaces Over a Finite Field (1977).- E. Calabi, Métriques kählériennes et fibrés holomorphes (1978).- E. Calabi, Isometric Families of Kähler Structures (1980).- E. Calabi, Géométrie différentielle affine des hypersurfaces (1981).- E. Calabi, Linear Systems of Real Quadratic Forms II (1982).- E. Calabi, Extremal Kähler Metrics (1982).- E. Calabi, Hypersurfaces with Maximal Affinely Invariant Area (1982).- E. Calabi, Extremal Kähler Metrics II (1985).- E. Calabi, Convex Affine Maximal Surfaces (1988).- E. Calabi, Affine Differential Geometry and Holomorphic Curves (1990).- E. Calabi, J. Cao Simple Closed Geodesics on Convex Surfaces (1992).- F. Beukers, J. A. C. Kolk and E. Calabi, Sums of Generalized Harmonic Series and Volumes (1993).- E. Calabi and H. Gluck, What are the Best Almost-Complex Structures on the 6-Sphere? (1993).- E. Calabi, Extremal Isosystolic Metrics for Compact Surfaces (1996).- E. Calabi, P. J. Olver, A. Tannenbaum, Affine Geometry, Curve Flows, and Invariant Numerical Approximations (1996).- J.-P. Bourguignon, E. Calabi, J. Eells, O. Garcia-Prada, M. Gromov, Where Does Geometry Go? A Research and Education Perspective (2001).- E. Calabi, X. Chen, The Space of Kähler Metrics II (2002).- Acknowledgements.

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Book SynopsisThe third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.Table of Contents Finite algebras and their clones Abstract clone theory Commutator theory Bibliography Index

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Book SynopsisThis 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 modeTrade ReviewWill be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field. * EMS *This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. * Zentralblatt MATH *A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. * Zentralblatt MATH *Table of ContentsIntroduction ; 1. Some topics in commutative algebra ; 2. General Properties of schemes ; 3. Morphisms and base change ; 4. Some local properties ; 5. Coherent sheaves and Cech cohmology ; 6. Sheaves of differentials ; 7. Divisors and applications to curves ; 8. Birational geometry of surfaces ; 9. Regular surfaces ; 10. Reduction of algebraic curves ; Bibilography ; Index

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Book SynopsisAn accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, a thorough treatment of Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields. Experts in the field will enjoy some of the new approaches to classical results.The text uses algebraic groups as the main examples, including worked out examples, instructive exercises, as well as bibliographical and historical remTrade Review'The author's intention was to write a quick introduction to the area of algebraic groups of the Lie type over fields of positive characteristic and I think he was very successful. The first part of the book can be recommended as a very suitable text for undergraduate students at the beginning of their studies.' * EMS Newsletter *The style of exposition in the book is very reader-friendly ... The proofs are clear and complete. * Mathematical Reviews *Table of Contents1. Algebraic sets and algebraic groups ; 2. Affine varieties and finite morphisms ; 3. Algebraic representations and Borel subgroups ; 4. Frobenius maps and finite groups of Lie type ; Bibliography ; Index

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Book SynopsisIn this modern treatment of the topic, Rolland Trapp presents an accessible introduction to the topic of multivariable calculus, supplemented by the use of fully interactive three-dimensional graphics throughout the text. Multivariable Calculus opens with an introduction to points, curves and surfaces, easing student transitions from two- to three-dimensions, and concludes with the main theorems of vector calculus. All standard topics of multivariable calculus are covered in between, including a variety of applications within the physical sciences. The exposition combines rigor and intuition, resulting in a well-rounded resource for students of the subject. In addition, the interactive three-dimensional graphics, accessible through the electronic text or via the companion website, enhance student understanding while improving their acuity. The style of composition, sequencing of subjects, and interactive graphics combine to form a useful text that appeals to a broad audience: studentsTrade Review[the book] covers the basic topics of calculus with functions of several variables with an original presentation that insists more on calculus and geometric interpretations than on theoretical details. It will certainly be very useful for students and for anyone looking for a first introduction to multivariable calculus. Reading it is a pleasure. * Julià Cufí, zbMath *Multivariable Calculus is, in my view, extremely well written and an all-round superb book from which students of mathematics and the physical sciences can gain an excellent understanding of the subject. The exposition is relaxed and informal, reading more like a conversation in parts...the choice of narrative style will, I imagine to a wide target audience and ensure as far as possible that the book remains accessible for all. * James Christian Cmath, FIMA, University of Salford, Mathematics Today *Table of Contents1: Introduction to three dimensions 2: Introduction to vectors 3: Differentiation 4: Integration 5: Vector Analysis

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Book SynopsisThis new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with 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. CastelnuovoTrade ReviewReview from previous edition Will be useful to graduate students as an introduction to arithmetic algebraic geometry, and to more advanced readers and experts in the field. * EMS *This book is unique in the current literature on algebraic and arithmetic geometry, therefore a highly welcome addition to it, and particularly suitable for readers who want to approach more specialized works in this field with more ease. The exposition is exceptionally lucid, rigorous, coherent and comprehensive. * Zentralblatt MATH *A thorough and far-reaching introduction to algebraic geometry in its scheme-theoretic setting ... The rich bibliography with nearly 100 references enhances the value of this textbook as a great introduction and source for research. * Zentralblatt MATH *Table of ContentsIntroduction ; 1. Some topics in commutative algebra ; 2. General Properties of schemes ; 3. Morphisms and base change ; 4. Some local properties ; 5. Coherent sheaves and Cech cohmology ; 6. Sheaves of differentials ; 7. Divisors and applications to curves ; 8. Birational geometry of surfaces ; 9. Regular surfaces ; 10. Reduction of algebraic curves ; Bibilography ; Index

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Book SynopsisThis textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection beTrade ReviewThe subject of the book is fascinating and written versions of the lecture series are nicley presented and preserve well the informal spirit of the lectures. This is a very useful book for graduate students and for mathematicians (or physicists) from other fields interested in the topic. * EMS *The lecturers cover an enormous amount of material, ranging from algeraic geometry and the theory of Riemann surfaces to loop groups, connections, Yang-Mills equations and twister theory. However despite this wide range, the book is surprisingly self-contained and readable. * Bulletin of the London Mathematical Society *Table of Contents1. Introduction ; 2. Riemann surfaces and integrable systems ; 3. Integrable systems and inverse scattering ; 4. Integrable systems and twistors ; Index

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Book SynopsisRecent developments are covered Contains over 100 figures and 250 exercises Includes complete proofsTrade ReviewFrom the reviews: "The book under review constitutes a self-contained introduction to the use of combinatorial methods in commutative algebra. … Concrete calculations and examples are used to introduce and develop concepts. Numerous exercises provide the opportunity to work through the material and end of chapter notes comment on the history and development of the subject. The authors have provided us with a useful reference and an effective text book." (R. J. Shank, Zentralblatt MATH, Vol. 1090 (16), 2006)Table of ContentsMonomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plücker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

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Book Synopsis1 From Congruent Numbers to Elliptic Curves.- II The HasseWeil L-Function of an Elliptic Curve.- III Modular forms.- IV Modular Forms of Half Integer Weight.- Answers, Hints, and References for Selected Exercises.Table of ContentsPreface to First Edition; Preface to Second Edition; Chapter I: From Congruent Numbers to Elliptic Curves; Chapter II: The Hasse-Weil L-Function of an Elliptic Curve; Chapter III: Modular Forms; Chapter IV Modular Forms of Half Integer Weight; Answers, Hints, and Exercises for Selected Exercises; Bibliography; Index

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Book SynopsisBasic Definitions.- Examples.- Projective Schemes.- Classical Constructions.- Local Constructions.- Schemes and Functors.Trade Review"A great subject and expert authors!"Nieuw Archief voor Wiskunde,June 2001"Both Eisenbud and Harris are experienced and compelling educators of modern mathematics. This book is strongly recommended to anyone who would like to know what schemes are all about."Newsletter of the New Zealand Mathematical Society, No. 82, August 2001Table of Contents1 Basic Definitions 2 Examples 3 Projective Schemes 4 Classical Constructions 5 Local Constructions 6 Schemes and Functors

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Book SynopsisBasic Definitions.- Examples.- Projective Schemes.- Classical Constructions.- Local Constructions.- Schemes and Functors.Trade Review"A great subject and expert authors!"Nieuw Archief voor Wiskunde,June 2001"Both Eisenbud and Harris are experienced and compelling educators of modern mathematics. This book is strongly recommended to anyone who would like to know what schemes are all about."Newsletter of the New Zealand Mathematical Society, No. 82, August 2001Table of Contents1 Basic Definitions 2 Examples 3 Projective Schemes 4 Classical Constructions 5 Local Constructions 6 Schemes and Functors

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Book SynopsisI An Overview of the Proof of Fermat's Last Theorem.- II A Survey of the Arithmetic Theory of Elliptic Curves.- III Modular Curves, Hecke Correspondences, and L-Functions.- IV Galois Coharnology.- V Finite Flat Group Schemes.- VI Three Lectures on the Modularity of% MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qacuaHbpGCgaqea8aadaWgaaWcbaWdbiaadweacaGGSaGaaG4maaWd% aeqaaaaa!3A7D!$${{\bar{\rho }}_{{E,3}}}$$and the Langlands Reciprocity Conjecture.- VII Serre's Conjectures.- VIII An Introduction to the Deformation Theory of Galois Representations.- IX Explicit Construction of Universal Deformation Rings.- X Hecke Algebras and the Gorenstein Property.- XI Criteria for Complete Intersections.- XII ?-adic Modular Deformationsand Wiles's Main Conjecture.- XIII The Flat Deformation Functor.- XIV Hecke Rings and Universal Deformation Rings.- XV Explicit Families of Elliptic Curves with Prescribed Mod NRepresentations.- XVI Modularity of Mod 5 Representations.- XVII An Extension of Wiles' Results.- Appendix to Chapter XVII Classification of% MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8% qacuaHbpGCgaqea8aadaWgaaWcbaWdbiaadweacaGGSaGaeS4eHWga% paqabaaaaa!3AF1!$${{\bar{\rho }}_{{E,\ell }}}$$by the jInvariant of E.- XVIII Class Field Theory and the First Case of Fermat's Last Theorem.- XIX Remarks on the History of Fermat's Last Theorem 1844 to 1984.- XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves.- XXI Wiles' Theorem andthe Arithmetic of Elliptic Curves.Trade Review"The story of Fermat's last theorem (FLT) and its resolution is now well known. It is now common knowledge that Frey had the original idea linking the modularity of elliptic curves and FLT, that Serre refined this intuition by formulating precise conjectures, that Ribet proved a part of Serre's conjectures, which enabled him to establish that modularity of semistable elliptic curves implies FLT, and that finally Wiles proved the modularity of semistable elliptic curves. The purpose of the book under review is to highlight and amplify these developments. As such, the book is indispensable to any student wanting to learn the finer details of the proof or any researcher wanting to extend the subject in a higher direction. Indeed, the subject is already expanding with the recent researches of Conrad, Darmon, Diamond, Skinner and others. ... FLT deserves a special place in the history of civilization. Because of its simplicity, it has tantalized amateurs and professionals alike, and its remarkable fecundity has led to the development of large areas of mathematics such as, in the last century, algebraic number theory, ring theory, algebraic geometry, and in this century, the theory of elliptic curves, representation theory, Iwasawa theory, formal groups, finite flat group schemes and deformation theory of Galois representations, to mention a few. It is as if some supermind planned it all and over the centuries had been developing diverse streams of thought only to have them fuse in a spectacular synthesis to resolve FLT. No single brain can claim expertise in all of the ideas that have gone into this "marvelous proof". In this age of specialization, where "each one of us knows more and more about less and less", it is vital for us to have an overview of the masterpiece such as the one provided by this book." (M. Ram Murty, Mathematical Reviews)Table of ContentsI An Overview of the Proof of Fermat’s Last Theorem.- II A Survey of the Arithmetic Theory of Elliptic Curves.- III Modular Curves, Hecke Correspondences, and L-Functions.- IV Galois Coharnology.- V Finite Flat Group Schemes.- VI Three Lectures on the Modularity of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacuaHbpGCgaqea8aadaWgaaWcbaWdbiaadweacaGGSaGaaG4maaWd % aeqaaaaa!3A7D! $${{\bar{\rho }}_{{E,3}}}$$ and the Langlands Reciprocity Conjecture.- VII Serre’s Conjectures.- VIII An Introduction to the Deformation Theory of Galois Representations.- IX Explicit Construction of Universal Deformation Rings.- X Hecke Algebras and the Gorenstein Property.- XI Criteria for Complete Intersections.- XII ?-adic Modular Deformations and Wiles’s “Main Conjecture”.- XIII The Flat Deformation Functor.- XIV Hecke Rings and Universal Deformation Rings.- XV Explicit Families of Elliptic Curves with Prescribed Mod NRepresentations.- XVI Modularity of Mod 5 Representations.- XVII An Extension of Wiles’ Results.- Appendix to Chapter XVII Classification of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacuaHbpGCgaqea8aadaWgaaWcbaWdbiaadweacaGGSaGaeS4eHWga % paqabaaaaa!3AF1! $${{\bar{\rho }}_{{E,\ell }}}$$ by the jInvariant of E.- XVIII Class Field Theory and the First Case of Fermat’s Last Theorem.- XIX Remarks on the History of Fermat’s Last Theorem 1844 to 1984.- XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves.- XXI Wiles’ Theorem and the Arithmetic of Elliptic Curves.

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Book SynopsisThis book provides the first comprehensive introduction to the circle of ideas developed around Mori's program, the prerequisites being only a basic knowledge of the algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.Trade ReviewReview of the hardback: '… the book is very crisply written, unusually easy to read for a book covering advanced material, and is moreover very concise for the book for reference, but is also an ideal book on which to base a series of seminars for research students, or indeed for research students to read by themselves.' P. M. H. Wilson, Bulletin of the London Mathematical SocietyReview of the hardback: '… a very good survey of present research.' European Mathematical SocietyReview of the hardback: 'I can recommend it to anyone wanting to get a deeper knowledge than just getting a survey of some facts on the classification theory.' M. Coppens, Niew Archief voor WiskundeReview of the hardback: '… a very good survey of present research … a very clear presentation of the subject.' EMSTable of Contents1. Rational curves and the canonical class; 2. Introduction to minimal model program; 3. Cone theorems; 4. Surface singularities; 5. Singularities of the minimal model program; 6. Three dimensional flops; 7. Semi-stable minimal models.

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Book SynopsisIn this introduction to commutative algebra, the author leads the beginning student through the essential ideas, without getting embroiled in technicalities. The route chosen takes the reader quickly to the fundamental concepts for understanding complex projective geometry, the only prerequisites being a basic knowledge of linear and multilinear algebra and some elementary group theory. In the first part, the general theory of Noetherian rings and modules is developed. A certain amount of homological algebra is included, and rings and modules of fractions are emphasised, as preparation for working with sheaves. In the second part, the central objects are polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalisation lemma and Hilbert's Nullstellensatz, affine complex schemes and their morphisms are introduced; Zariski's main theorem and Chevalley's semi-continuity theorem are then proved. Finally, a detailed study of Weil and CartTrade Review'… a detailed study … a solid background.' L'Enseignement MathématiqueTable of Contents1. Rings, homomorphisms, ideals; 2. Modules; 3. Noetherian rings and modules; 4. Artinian rings and modules; 5. Finitely generated modules over Noetherian rings; 6. A first contact with homological algebra; 7. Fractions; 8. Integral extensions of rings; 9. Algebraic extensions of rings; 10. Noether's normalisation lemma; 11. Affine schemes; 12. Morphisms of affine schemes; 13. Zariski's main theorem; 14. Integrally closed Noetherian rings; 15. Weil divisors; 16. Cartier divisors; Subject index; Symbols index.

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Book SynopsisThis edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.Trade Review'The authors have created a true masterpiece of mathematical exposition. Bringing together disparate ideas developed gradually over the last fifty years into a cohesive whole, Huybrechts and Lehn provide a compelling and comprehensive view of an essential topic in algebraic geometry. The new edition is full of gems that have been discovered since the first edition. This inspiring book belongs in the hands of any mathematician who has ever encountered a vector bundle on an algebraic variety.' Max Lieblich, University of Washington'This book fills a great need: it is almost the only place the foundations of the moduli theory of sheaves on algebraic varieties appears in any kind of expository form. The material is of basic importance to many further developments: Donaldson–Thomas theory, mirror symmetry, and the study of derived categories.' Rahul Pandharipande, Princeton University'This is a wonderful book; it's about time it was available again. It is the definitive reference for the important topics of vector bundles, coherent sheaves, moduli spaces and geometric invariant theory; perfect as both an introduction to these subjects for beginners, and as a reference book for experts. Thorough but concise, well written and accurate, it is already a minor modern classic. The new edition brings the presentation up to date with discussions of more recent developments in the area.' Richard Thomas, Imperial College London'Serving as a perfect introduction for beginners in the field, an excellent guide to the forefront of research in various directions, a valuable reference for active researchers, and as an abundant source of inspiration for mathematicians and physicists likewise, this book will certainly maintain both its particular significance and its indispensability for further generations of researchers in the field of algebraic sheaves (or vector bundles) and their moduli spaces.' Zentralblatt MATHTable of ContentsPreface to the second edition; Preface to the first edition; Introduction; Part I. General Theory: 1. Preliminaries; 2. Families of sheaves; 3. The Grauert–Müllich Theorem; 4. Moduli spaces; Part II. Sheaves on Surfaces: 5. Construction methods; 6. Moduli spaces on K3 surfaces; 7. Restriction of sheaves to curves; 8. Line bundles on the moduli space; 9. Irreducibility and smoothness; 10. Symplectic structures; 11. Birational properties; Glossary of notations; References; Index.

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Book SynopsisMatroid theory is a vibrant area of research that provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This book provides the first comprehensive introduction to the field which will appeal to undergraduate students and to any mathematician interested in the geometric approach to matroids. Written in a friendly, fun-to-read style and developed from the authors' own undergraduate courses, the book is ideal for students. Beginning with a basic introduction to matroids, the book quickly familiarizes the reader with the breadth of the subject, and specific examples are used to illustrate the theory and to help students see matroids as more than just generalizations of graphs. Over 300 exercises are included, with many hints and solutions so students can test their understanding of the materials covered. The authors have also included several projects and open-ended research problems for independent study.Trade Review"The authors write in an entertaining, conversational style, and the text is often peppered with humorous footnotes. Nearly 300 exercises and scores of references will benefit motivated readers." -J. T. Saccoman, ChoiceTable of Contents1. A tour of matroids; 2. Cryptomorphisms; 3. New matroids from old; 4. Graphic matroids; 5. Finite geometry; 6. Representable matroids; 7. Other matroids; 8. Matroid minors; 9. The Tutte polynomial; Projects; Appendix: matroid axiom systems; Bibliography; Index.

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