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Algebra, Geometry, Topology, and Functional Analysis Equations of Major Unsolved Problems in Mathematics

(Hodge Conjecture). Description: Let X be a smooth projective complexalgebraic variety. Its cohomology admits a Hodge decomposition:H2k(X, C) = Mp+q=2kHp,q(X).Conjecture: Every class inHk,k(X) ∩ H2k(X, Q)is a rational linear combination of classes of algebraic cycles.2Problem 1.2 (Jacobian Conjecture). Description: Let F = (F1, . . . , Fn) : Cn → Cn be
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