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In complex analysis and algebraic geometry one studies complex analytic and algebraic varieties, maps between such spaces (the simplest case being holomorphic and algebraic functions) and analytic and algebraic objects defined on those spaces, as subvarieties, vector bundles and sheaves. There are many relations of complex analysis and algebraic geometry to other fields of mathematics, for example functional analysis, algebraic topoplogy and commutative algebra. A classical application of complex analysis is analytic number theory. In recent years elliptic curves, a favourite theme of study in complex analysis and algebraic geometry, have become an important tool in algorithmic number theory and in cryptography. Various parts of complex analysis and algebraic geometry (e.g. deformation theory and the theory of moduli spaces) have become relevant for theoretical physics.