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14: Algebraic geometry


Introduction

Algebraic geometry combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" -- ostensibly a statement in number theory -- was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves.

History

See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)

Applications and related fields

[Schematic of subareas and related areas]

Subfields

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

Textbooks: Hartshorn; ...

For a more elementary introduction see Reid, Miles: "Undergraduate algebraic geometry", London Mathematical Society Student Texts, 12. Cambridge University Press, Cambridge-New York, 1988. 129 pp. ISBN 0-521-35559-1; 0-521-35662-8 MR90a:14001

Some survey articles:

Software and tables

Note that many computations in algebraic geometry are really computations in polynomials rings, hence computational commutative algebra applies.

Other web sites with this focus

Selected topics at this site


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Last modified 2000/01/14 by Dave Rusin. Mail: