Gradient conjecture

In mathematics, the gradient conjecture, due to René Thom (1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (University of Savoie, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusiński (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 at x₀, 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 during that period by Thom during his weekly seminar on singularities at the IHES.)
• The paper where it is proved: Annals of Mathematics 152 (2000), 763-792. It is available here.