Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Notes and history

Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Enriques classified algebraic surfaces up to birational isomorphism. The style of the Italian school was very intuitive and does not meet the modern standards of rigor.

Related Topics:
Italian geometers - Algebraic surfaces - Birational isomorphism - Rigor

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By the 1930s and 1940s, Oscar Zariski, André 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. For a while there was no standard foundation for algebraic geometry.

Related Topics:
1930s - 1940s - Oscar Zariski - André Weil - Commutative algebra - Valuation theory - Elimination theory - Ideal theory - Modules - David Hilbert - Max Noether - Emanuel Lasker - Emmy Noether - Wolfgang Krull

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In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of the theory of sheaf theory. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli.

Related Topics:
1950s - 1960s - Jean-Pierre Serre - Alexander Grothendieck - Sheaf theory - Schemes - Homological techniques - 1970s - Number theory - Singularities - Moduli

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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.

Related Topics:
Abelian varieties - Group

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The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography.

Related Topics:
Elliptic curve - Fermat's last theorem - Elliptic curve cryptography

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While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems.

Related Topics:
Gröbner bases - Computer algebra

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