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

(Hodge Conjecture). Description: Let X be a smooth projective complex
algebraic variety. Its cohomology admits a Hodge decomposition:
H2k(X, C) = M
p+q=2k
Hp,q(X).
Conjecture: Every class in
Hk,k(X) ∩ H2k(X, Q)
is a rational linear combination of classes of algebraic cycles.
2
Problem 1.2 (Jacobian Conjecture). Description: Let F = (F1, . . . , Fn) : Cn → Cn be a
polynomial map such that its Jacobian determinant,
JF (x) = det
∂Fi
∂xj

,
is a nonzero constant. Prove that F is invertible and that its inverse is also given by a
polynomial map.
Problem 1.3 (Smooth 4-Dimensional Poincaré Conjecture). Description: Prove or refute
that any smooth, compact 4-manifold without boundary that is homotopy equivalent to the
4-sphere S4 is diffeomorphic to S4.
Problem 1.4 (Invariant Subspace Problem). Description: Let T be a bounded linear
operator on an infinite-dimensional, separable Hilbert space H. Determine whether there
always exists a closed, nontrivial subspace M ⊂ H (i.e., {0}̸ = M̸ = H) such that:
T (M ) ⊆ M.