**Inserted:** 1 apr 1996

**Year:** 1995

**Notes:**

Unpublished. Translated in English by G. Alberti, L. Ambrosio, C. Mantegazza, V. M. Tortorelli

**Abstract:**

Some conjectures concerning ``evolution problems'' are presented. They are related to the ``steepest descent'', to the approximations of Newton's gravitation law, to hyperbolic non linear equations and to ``descent movements'' of manifolds. The article identifies several questions, of which I do not know the solution, and points out a number of analogies between problems that are apparently far from each other. I believe that the study of these conjectures might provide an opportunity for scientists that are expert in different fields within pure and applied mathematics to get together and ponder on the connections that exist among various mathematical concepts, such as linear vs. non-linear behavior, stability vs. instability, or convergence of different approximation methods, and certain ideas well developed in physics, such as deterministic vs. non-deterministic behavior, predictability vs. non-predictability, order and chaos, etc.

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