mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the gradient conjecture, due to

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became w ...

(1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka (

University of Savoie Savoy Mont Blanc University (french: Université Savoie Mont Blanc, a.k.a. Chambéry University) is a public university in the region of Savoy, with one campus in Annecy and two around Chambéry. Campuses The university was officially founded i ...

, France), Tadeusz Mostowski ( Warsaw University, Poland) and Adam Parusiński ( University of Angers, France). The conjecture states that given a real-valued analytic function ''f'' defined on R^''n'' and a trajectory ''x''(''t'') of the gradient vector field of ''f'' having a limit point ''x''₀ ∈ R^''n'', where ''f'' has an isolated critical point at ''x''₀, there exists a limit (in the

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

PR^''n-1'') for the secant lines from ''x''(''t'') to ''x''₀, as ''t'' tends to zero. The proof depends on a theorem due to

Stanis%C5%82aw %C5%81ojasiewicz Stanisław Łojasiewicz (9 October 1926 – 14 November 2002) was a Polish mathematician.. Biography At the end of the 1950s, he solved the problem of distribution division by analytic functions, introducing the %C5%81ojasiewicz inequality. Its ...

References

* R. Thom (1989) "Problèmes rencontrés dans mon parcours mathématique: un bilan", Publications Math%C3%A9matiques de l%27IH%C3%89S 70: 200 to 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.) * In 2000 the conjecture was proven correct in Annals of Mathematics 152: 763 to 792. The proof is availabl
here
Theorems in analysis {{Mathanalysis-stub