Algebraic Geometry is a subject with historical roots in analytic geometry. At its most naive level it is concerned with the geometry of the solutions of a system of polynomial equations. In its early days the subject developed around the classification problem, the search for invariants of transformations, intersection problems, and the study of families of points on a curve or curves on a surface (known as linear systems). It made use of techniques from geometry (projective geometry), number theory (Diophantine equations), and analysis (elliptic and abelian integrals). Today it is a powerful synthesis of those algebraic, geometric, and analytic techniques. Results can be universally applied to a range of problems from the discrete (such as the recent proof of Fermat's Last Theorem) to the continuous (global complex analysis). It subsumes most of commutative algebra and much of algebraic number theory, and overlaps with differential geometry, modern "analytic geometry" (complex manifolds), Lie groups, representation theory, theoretical physics, and to a lesser extent the theory of partial differential equations. In addition to being one of the central disciplines of pure mathematics, algebraic geometry has developed an applied side which is linked to problems in computational complexity and the theory of algorithms, symbolic computation, robotics, control theory, computational geometry, geometric modeling, image recognition, computer vision, and scientific visualization.
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