Gradient conjecture
Encyclopedia
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 at x0, there exists a limit (in the projective space
PRn-1) for the secant
lines from x(t) to x0, as t tends to zero.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the gradient conjecture, due to René Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
, was proved in 2000 by 3 Polish mathematicians, Krzysztof Kurdyka (University of Savoie
University of Savoie
The University of Savoy is a university in alpine eastern France, in the historical region of Savoy, with one campus in Annecy-le-Vieux and two around Chambéry .-Campuses:...
, France), Tadeusz Mostowski (Warsaw University, Poland) and Adam Parusinski (University of Angers
University of Angers
The University of Angers is an institution of higher learning situated in the town of the same name, in western France. It was founded in 1356, closed down in 1793, and reestablished in 1971....
, France). It states that given a real-valued analytic function
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, categories that are similar in some ways, but different in others...
f defined on Rn and a trajectory
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
x(t) of the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
vector field of f having a limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
x0 ∈ Rn, where f has an isolated critical point at x0, there exists a limit (in the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
PRn-1) for the secant
Secant
Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the secant variety, in algebraic geometry...
lines from x(t) to x0, as t tends to zero.