The gradient conjecture, due to René Thom, was proved in 2000 by K. Kurdyka, T. Mostowski and A. Parusinski. It states that given an analytic function f in Rⁿ and a trajectory x(t) of the gradient vector field of f having a limit point x₀ ∈ Rⁿ, there exists a limit (in the projective space PRⁿ) for the secants of x(t) near x₀.

References

• The original statement: R. Thom, Problèmes rencontrés dans mon parcours mathématique: un bilan, Publ. Math. IHES 70 (1989), 200-214.
• The paper where it is proved: Annals of Math. 152 (2000), 763-792. It is available here.