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What (who) is algebraic geometry - definition
Algebraic geometry
BRANCH OF MATHEMATICS DEALING WITH ALGEBRAIC VARIETIES AND THEIR GENERALIZATIONS (SCHEMES, ETC.)
Algebraic Geometry; Computational algebraic geometry; History of algebraic geometry; Applications of algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
Derived algebraic geometry
BRANCH OF MATHEMATICS GENERALIZING ALGEBRAIC GEOMETRY SO THAT COMMUTATIVE RINGS PROVIDING LOCAL CHARTS ARE REPLACED BY SIMPLICIAL COMMUTATIVE RINGS OR E∞-RING SPECTRA, WHOSE HIGHER HOMOTOPY GROUPS ACCOUNT FOR NON-DISCRETENESS OF THE STRUCTURE SHEAF
Homotopical algebraic geometry; Spectral algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb{Q}), simplicial commutative rings or E_{\infty}-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g.
Arithmetic geometry
![The [[hyperelliptic curve]] defined by <math>y^2=x(x+1)(x-3)(x+2)(x-2)</math> has only finitely many [[rational point]]s (such as the points <math>(-2, 0)</math> and <math>(-1, 0)</math>) by [[Faltings's theorem]].](https://commons.wikimedia.org/wiki/Special:FilePath/Example of a hyperelliptic curve.svg?width=200 "The [[hyperelliptic curve]] defined by <math>y^2=x(x+1)(x-3)(x+2)(x-2)</math> has only finitely many [[rational point]]s (such as the points <math>(-2, 0)</math> and <math>(-1, 0)</math>) by [[Faltings's theorem]].")
BRANCH OF ALGEBRAIC GEOMETRY FOCUSED ON PROBLEMS IN NUMBER THEORY
Arithmetical algebraic geometry; Arithmetic Geometry; Arithmetic algebraic geometry; Arithmetic Algebraic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
Examples of use of algebraic geometry
1. US mathematician David Mumford, a professor at Brown University’s Applied Mathematics Division, was co–winner of the Wolf Prize on Sunday for his groundbreaking theoretical work in algebraic geometry.