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Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they are isomorphic (that is, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for the problem.
Information bottleneck method; Inverse chain rule method ; Inverse transform sampling method (probability) Iterative method (numerical analysis) Jacobi method (linear algebra) Largest remainder method (voting systems) Level-set method; Linear combination of atomic orbitals molecular orbital method (molecular orbitals) Method of characteristics
Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space). Geometric calculus , an extension of GA that incorporates differentiation and integration , can be used to formulate other theories such as complex analysis and differential geometry , e.g. by using the Clifford ...
Algebraic variety. Hypersurface; Quadric (algebraic geometry) Dimension of an algebraic variety; Hilbert's Nullstellensatz; Complete variety; Elimination theory; Gröbner basis; Projective variety; Quasiprojective variety; Canonical bundle; Complete intersection; Serre duality; Spaltenstein variety; Arithmetic genus, geometric genus, irregularity
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Abhyankar–Moh theorem (algebraic geometry) Addition theorem (algebraic geometry) Andreotti–Frankel theorem (algebraic geometry) Arithmetic Riemann–Roch theorem (algebraic geometry) BBD decomposition theorem (algebraic geometry) Base change theorems (algebraic geometry) Beauville–Laszlo theorem (vector bundles) Belyi's theorem (algebraic ...
Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2).