Lesson plan
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). 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 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.
Article summary
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). 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 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.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.