mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 ...

(1989), was proved in 2000 by three Polish mathematicians, Krzysztof Kurdyka ( University of Savoie, France), Tadeusz Mostowski (

Warsaw University The University of Warsaw (, ) is a public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well as 100 specializat ...

, Poland) and Adam Parusiński (

University of Angers The University of Angers () is a public university in western France, with campuses in Angers, Cholet, and Saumur. It is part of the Angers-Le Mans University Community. History The University of Angers was initially established during the 11 ...

, France). The conjecture states that given a real-valued

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...

''f'' defined on R^''n'' and a

trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...

''x''(''t'') of the

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

vector field of ''f'' having a

limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...

''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 line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inter ...

s 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 ''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recher ...

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 The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

152: 763 to 792. The proof is availabl
here
Theorems in mathematical analysis {{Mathanalysis-stub