Gradient conjecture

In mathematics, the gradient conjecture, due to René Thom, was proved in 2000 by 3 Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusinski (University of Angers, France). It states that given a real-valued analytic function f defined on Rⁿ and a trajectory x(t) of the gradient vector field of f having a limit point x₀ ∈ Rⁿ, where f has an isolated critical point, there exists a limit (in the projective space PR^n-1) for the secant lines from x(t) to x₀, as t tends to zero.

References

• A published statement of the conjecture: R. Thom, Problèmes rencontrés dans mon parcours mathématique: un bilan, Publ. Math. IHES 70 (1989), 200-214. (This gradient conjecture due to René Thom was in fact well-known among specialists by the early 70's, having been often discussed from that period by Thom during his weekly seminar on singularities at the IHES.)
• The paper where it is proved: Annals of Math. 152 (2000), 763-792. It is available here.