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This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes. Prevarieties - Spectrum of a Ring - Schemes - Fiber products - Schemes over fields - Local properties of schemes - Quasi-coherent modules - Representable functors - Separated morphisms - Finiteness Conditions - Vector bundles - Affine and proper morphisms - Projective morphisms - Flat morphisms and dimension - One-dimensional schemes - Examples Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.

Mathematics. --- Algebra. --- Mathématiques --- Algèbre --- Géométrie algébrique --- Geometric geometry. --- Geometry, algebraic. --- Algebraic Geometry. --- Mathematics --- Mathematical analysis --- Algebraic geometry --- Geometry --- Geometry, Algebraic. --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Algebraic geometry.

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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.

Geometry, Algebraic --- Geometry, Analytic --- Géométrie algébrique --- Géométrie analytique --- Géométrie algébrique --- Géométrie analytique --- Geometry, Algebraic. --- Geometry, Analytic. --- Analytical geometry --- Algebra --- Conic sections --- Algebraic geometry --- Geometry --- Graphic methods

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In recent years, the discovery of new algorithms for dealing with polynomial equations, coupled with their implementation on fast inexpensive computers, has sparked a minor revolution in the study and practice of algebraic geometry. These algorithmic methods have also given rise to some exciting new applications of algebraic geometry. This book illustrates the many uses of algebraic geometry, highlighting some of the more recent applications of Gröbner bases and resultants. The book is written for nonspecialists and for readers with a diverse range of backgrounds. It assumes knowledge of the material covered in a standard undergraduate course in abstract algebra, and it would help to have some previous exposure to Gröbner bases. The book does not assume the reader is familiar with more advanced concepts such as modules. For the new edition, the authors have added a unified discussion of how matrices can be used to specify monomial orders; a revised presentation of the Mora normal form algorithm; two sections discussing the Gröbner fan of an ideal and the Gröbner Walk basis conversion algorithm; and a new chapter on the theory of order domains, associated codes, and the Berlekamp-Massey-Sakata decoding algorithm. They have also updated the references, improved some of the proofs, and corrected typographical errors. David Cox is Professor of Mathematics at Amherst College. John Little is Professor of Mathematics at College of the Holy Cross. Dona l O’Shea is the Elizabeth T. Kennan Professor of Mathematics and Dean of Faculty at Mt. Holyoke College. These authors also co-wrote the immensely successful book, Ideals, Varieties, and Algorithms.

Geometry, Algebraic. --- Géométrie algébrique --- Geometry, Algebraic --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Géométrie algébrique --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Algebraic geometry --- Mathematics. --- Computer science --- Algebraic geometry. --- Algorithms. --- Algebraic Geometry. --- Symbolic and Algebraic Manipulation. --- Geometry, algebraic. --- Algebra --- Data processing. --- Algorism --- Arithmetic --- Foundations --- Computer science—Mathematics. --- Computer mathematics --- Electronic data processing

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The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology. Some topologists seek to understand geometric properties of solutions to finite systems of equations or inequalities and configurations which in some sense actually occur in the real world. Others study spaces constructed more abstractly using infinite limit processes. Their goal is to determine just how similar or different these abstract spaces are from those which are finitely described. However, as topology is usually taught, even the first, more concrete type of problem is approached using the language and methods of the second type. Professor Brumfiel's thesis is that this is unnecessary and, in fact, misleading philosophically. He develops a type of algebra, partially ordered rings, in which it makes sense to talk about solutions of equations and inequalities and to compare geometrically the resulting spaces. The importance of this approach is primarily that it clarifies the sort of geometrical questions one wants to ask and answer about those spaces which might have physical significance.

Categories (Mathematics) --- Commutative rings. --- Rings (Algebra) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Commutative rings --- 512.7 --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Categories (Mathematics). --- Ordered algebraic structures --- Algebraic geometry --- Anneaux commutatifs --- Catégories (Mathématiques) --- Anneaux (algèbre) --- Géométrie algébrique --- Structures algébriques ordonnées

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Lunar Outpost provides a detailed account of the various technologies, mission architectures, medical requirements and training needed to return humans to the Moon within the next decade. It focuses on the means by which a lunar outpost will be constructed and also addresses major topics such as the cost of the enterprise and the roles played by private companies and individual countries. The return of humans to the surface of the Moon will be critical to the exploration of the solar system. The various missions are not only in pursuit of scientific knowledge, but also looking to extend human civilization, economic expansion, and public engagement beyond Earth. As well as NASA, China’s Project 921, Japan’s Aerospace Exploration Agency, Russia, and the European Space Agency are all planning manned missions to the Moon and, eventually, to Mars. The Ares-I and Ares-V are the biggest rockets since the Saturn V and there is much state-of-the-art technology incorporated into the design of Orion, the spacecraft that will carry a crew of four astronauts to the Moon. Lunar Outpost also describes the human factors, communications, exploration activities, and life support constraints of the missions.

Space flight to the moon. --- Surfaces. --- Lunar bases --- Lunar excursion module --- Lunar probes --- Human settlements --- Aeronautics Engineering & Astronautics --- Mechanical Engineering --- Engineering & Applied Sciences --- Lunar bases. --- Lunar surface vehicles. --- Cars, Lunar --- Cars, Moon --- Lunar cars --- Lunar rovers --- Lunar roving vehicles --- Moon cars --- Surface vehicles, Lunar --- Lunar construction engineering --- Moon bases --- Moon settlements --- Flight to the moon --- Lunar expeditions --- Lunar flight --- Algebraic fields. --- Engineering. --- Astronomy. --- Aerospace engineering. --- Astronautics. --- Aerospace Technology and Astronautics. --- Popular Science in Astronomy. --- 51 --- 51 Mathematics --- Mathematics --- Geometry --- Ordered algebraic structures --- Space sciences --- Aeronautics --- Astrodynamics --- Space flight --- Space vehicles --- Aeronautical engineering --- Astronautics --- Engineering --- Physical sciences --- Construction --- Industrial arts --- Technology --- Extraterrestrial bases --- Roving vehicles (Astronautics) --- Associative rings. --- K-theory. --- Polyhedra --- Singularities (Mathematics) --- Surfaces --- Congresses. --- Corps algébriques. --- Geometry, Algebraic --- Géométrie algébrique --- Singularités (mathématiques) --- Géométrie algébrique. --- Algèbres associatives --- Moon --- Exploration.