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# Topic: Algebraic geometry

 PlanetMath: algebraic geometry Algebraic geometry is the study of algebraic objects using geometrical tools. Algebraic groups are essentially matrix group schemes, and as such allow the tools of algebraic geometry to be applied to their study. This is version 11 of algebraic geometry, born on 2004-03-19, modified 2005-02-26. planetmath.org /encyclopedia/AlgebraicGeometry.html   (2500 words)

 Encyclopedia: Algebraic geometry Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations. Algebraic geometry was developed largely by the Italian geometers in the early part of the 20-th century. www.nationmaster.com /encyclopedia/Algebraic-geometry   (417 words)

 Arithmetic Algebraic Geometry However, developments in recent years have transformed the subject into one of the central areas of mathematical research, which has relations with, or applications to, virtually every mathematical field, as well as an impact to contemporary everyday life (for example, the use of prime numbers and factorisation for encoding "smart" cards). Through the interaction of arithmetic and geometry, these different theories have led to a complex and far-reaching web of conjectures proposing a deep explanation for the observed phenomena. In this framework, it is offering post- and pre-doctoral positions at each of its associated nodes, as part of its initiative to foster research in central subjects of this active domain of mathematical research. www.arithgeom-network.univ-rennes1.fr   (476 words)

 Geometry - Wikipedia, the free encyclopedia The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665). Developments in algebraic geometry included the study of curves and surfaces over finite fields, rather than the real or complex numbers. en.wikipedia.org /wiki/Geometry   (2280 words)

 AllRefer.com - algebraic geometry (Mathematics) - Encyclopedia algebraic geometry, branch of geometry, based on analytic geometry, that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates). In plane geometry an algebraic curve is the locus of all points satisfying the polynomial equation f(x,y)=0; in three dimensions the polynomial equation f(x,y,z)=0 defines an algebraic surface. The intersection of two or more algebraic hypersurfaces defines an algebraic set, or variety, a concept of particular importance in algebraic geometry. reference.allrefer.com /encyclopedia/A/algebGeo.html   (235 words)

 Algebraic geometry - Wikipedia, the free encyclopedia By the 1930s and 1940s, Oscar Zariski, AndrĂ© Weil and others realized that algebraic geometry needed to be rebuilt on foundations of commutative algebra and valuation theory. Commutative algebra (earlier known as elimination theory and then ideal theory, and refounded as the study of commutative rings and their modules) had been and was being developed by David Hilbert, Max Noether, Emanuel Lasker, Emmy Noether, Wolfgang Krull, and others. An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. en.wikipedia.org /wiki/Algebraic_geometry   (1810 words)

 Learn more about Algebraic geometry in the online encyclopedia.   (Site not responding. Last check: 2007-10-27) This correspondence of irreducible varieties and prime ideals is a central theme of algebraic geometry. Their work on birational geometry was deep; but didn't rest on a sufficiently rigorous basis. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20h century, with the geometric applications in mind. www.onlineencyclopedia.org /a/al/algebraic_geometry.html   (803 words)

 3.2.1 Algebraic Geometry   (Site not responding. Last check: 2007-10-27) Algebraic geometry is the study of algebraic varieties (for example, polynomial equations and their graphs). The classification of algebraic varieties, the geometry of special algebraic varieties and the description of mappings of algebraic varieties are important problems of current research interest. Algebraic geometry is closely related to differential geometry and topology, commutative algebra, and number theory. www.math.okstate.edu /grad/brief-hbk/3_2_1Algebraic_Geometry.html   (183 words)

 [No title] Algebraic geometry is one of the oldest and vastest branches of mathematics. In McGill, algebraic geometry is represented by Peter Russell and Eyal Goren and is very much connected to the interests of other members of the department: Henri Darmon, Jacques Husrtubise and Niky Kamran. The objects of the geometry are the prime ideals of the ring and the notion of "lying on" is given by "containing". www.math.mcgill.ca /goren/AGwebpage/webpage2.htm   (1584 words)

 14: Algebraic geometry Algebraic geometry combines the algebraic with the geometric for the benefit of both. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. Singular Package: Singular is a computer algebra system for singularity theory and algebraic geometry developed by G.-M. Greuel, G. Pfister, H. www.math.niu.edu /~rusin/known-math/index/14-XX.html   (523 words)

 Algebraic geometry   (Site not responding. Last check: 2007-10-27) Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially the study of commutative rings, and geometry. In algebraic geometry, geometric structures are defined as the set of zeros of a number of polynomials. Commutative algebra (as the study of commutative rings and their ideals) was developed by David Hilbert, Emmy Noether and others, also in the 20-th century, with the geometric applications in mind. brandt.kurowski.net /projects/lsa/wiki/view.cgi?doc=390   (613 words)

 Algebraic Geometry Algebraic Geometry is a new emphasis area within the department. Algebraic Geometry in simplest terms is the study of polynomial equations and the geometry of their solutions. The document Graduate Studies in Algebraic Geometry outlines the general areas of Algebraic Geometry studied here and describes the advanced undergraduate and graduate courses that are under development or offered regularly. www.math.uiuc.edu /ResearchAreas/AlgebraicGeometry   (311 words)

 Algebraic Geometry   (Site not responding. Last check: 2007-10-27) Algebraic geometry is the study of the "shape" of the set of solutions to polynomial equations. The study of this type of question is called "arithmetic algebraic geometry" and is closely related to number theory, group theory, and representation theory. The study of the geometric properties of this continuum is known as "complex algebraic geometry" and is closely related to topology, differential geometry, complex analysis and even theoretical physics. www.math.utah.edu /research/ag   (382 words)

 Amazon.com: An Invitation to Algebraic Geometry: Books   (Site not responding. Last check: 2007-10-27) The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Describes the underlying principles of algebraic geometry, for the working mathematician who is unfamiliar with algebraic geometry but wishes to learn about its foundations and its goals. www.amazon.com /exec/obidos/tg/detail/-/0387989803?v=glance   (1236 words)

 Research in Algebraic Geometry Algebraic varieties are the solution sets to systems of polynomial equations. Much of the work in the field of algebraic cycles is organized around three major conjectures: the Hodge conjecture, the Tate conjecture and the generalized Birch-Swinnerton Dyer conjecture. Algebraic Geometry and theoretical physics have found common interest in the study of "Calabi-Yau Manifolds". www.math.duke.edu /~schoen/researchalggeo.html   (712 words)

 The Math Forum - Math Library - Algebraic Geom.   (Site not responding. Last check: 2007-10-27) Algebraic Geometry preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives. An area of algebraic geometry that deals with nonsingular curves of genus 1 - in English, solutions to equations y^2 = x^3 + A x + B. It has important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography). An introduction to computational algebraic geometry and commutative algebra at the undergraduate level, with discussions of systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms"). mathforum.org /library/topics/algebraic_g   (1925 words)

 Algebraic geometry at opensource encyclopedia   (Site not responding. Last check: 2007-10-27) A algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. A regular function on an algebraic set V contained in $\left\{\mathbb A\right\}^n$ is defined to be the restriction of a regular function on $\left\{\mathbb A\right\}^n$, in the sense we defined above. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. wiki.tatet.org /Algebraic_geometry.html   (1720 words)

 IMA Thematic Year on Applications of Algebraic Geometry, September 2006 - June 2007 Algebraic geometry has a long and distinguished presence in the history of mathematics that produced both powerful and elegant theorems. In recent years new algorithms have been developed and several old and new methods from algebraic geometry have led to significant and unexpected advances in several diverse areas of application. Motivated by these exciting developments, the year in algebraic geometry and its applications aims to bring together mathematicians, computer scientists, economists, statisticians and engineers from various disciplines in order to enhance interactions, generate new applications and motivate further progress. www.ima.umn.edu /AlgGeom   (278 words)

 Amazon.ca: Algebraic Geometry: Books   (Site not responding. Last check: 2007-10-27) Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. www.amazon.ca /exec/obidos/ASIN/0387902449   (2022 words)

 Algebraic Geometry Conferences NATO Advanced Study Institute: The Arithmetic and Geometry of Algebraic Cycles, Banff Centre for Conferences, Banff, Alberta, Canada, 7-19 June 1998. Workshop on Diophantine Geometry Related to the ABC Conjecture, The University of Arizona, Tucson AZ, 14-18 March 1998. Midwest Algebraic Geometry Conference, The University of Notre Dame, 7-9 November 1997. www.cgtp.duke.edu /~drm/conferences/alg-geom.html   (334 words)

 ALGEBRAIC GEOMETRY This is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with a selection of topics from the theory of algebraic curves, including the Riemann—Roch theorem, elliptic curves, the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves. www.worldscibooks.com /mathematics/3873.html   (159 words)

 Algebraic Geometry   (Site not responding. Last check: 2007-10-27) The aim of this course is to provide an introduction to algebraic geometry at the beginning graduate level. Prerequisites are previous or concurrent enrollment in a graduate algebra course (Math 671-2). For more information about algebraic geometry in general, look at my list of resources for algebraic geometers. www.math.byu.edu /People/links/jarvis/algebra/alg-geom.html   (85 words)

 Introduction to Algebraic Geometry Here's a proof of Riemann-Roch and Serre duality (for curves) that I gave in the Baby Algebraic Geometry Seminar (dvi, ps, or pdf) (Feb. 11), that fits well at the end of these notes. The notes to Olivier Debarre's introductory course in algebraic geometry are available from his homepage (in french). The notes to Igor Dolgachev's introductory course in algebraic geometry are available from his lecture notes page. math.stanford.edu /~vakil/725/course.html   (983 words)

 School of Mathematics - Algebraic Geometry New Connections of Representation Theory to Algebraic Geometry and Physics During the academic year 2006-07, the School of Mathematics will have a special program on algebraic geometry. We don't want to focus on any single aspect, but rather aim to have many flavors of algebraic geometry and its applications represented, including (not exhaustive list) cohomology theories, motives, moduli spaces, Shimura varieties, complex or p-adic analytic methods and singularities. math.ias.edu /pages/activities/special-programs/algebraic-geometry.php   (167 words)

 University of Chicago Algebraic Geometry Seminar   (Site not responding. Last check: 2007-10-27) The algebraic geometry seminar is held in Eckhart Hall room 203, on Wednesdays at 4-5pm, unless otherwise specified. One of the fundamental invariants of the moduli space of curves is its cone of effective divisors which determines to a large extent all the rational maps from the moduli space to other varieties. We discuss a new equivalence relation, denoted by A.Q.E.D. (= Algebraic-Quasi-Etale-Deformation) for complete algebraic varieties with canonical singularities: it is generated by birational equivalence, by flat algebraic deformations, and by quasi-etale morphisms, i.e., morphisms which are unramified in codimension \$1\$. www.math.uchicago.edu /~howard/agcal.cgi   (788 words)

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