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V.I. Arnold 10 February, 1998

Summary: Problems
V.I. Arnold
10 February, 1998
1. Study from the singularity theory point of view the E.Cartan's theory
of pfaĈan systems. What is happening in the nonanalytical (smooth) case,
a) generically b) outside a "neglectable" variety of codimension in nity?
Example: nd the smooth version of the recent theory (by I.Ekeland)
of the disaggregation of the analytical demand function in economics.
] For a model problem (convex Darboux theorem) the smooth version
was provided by Zakalyukin.]
2. Consider a small generic smooth perturbation of RP 2 in RP 3 . A special
point is a point of the parabolic curve of the perturbed surface, where the
asymptotical direction is tangent to the parabolic curve (it corresponds to
a swallowtail on the dual surface). a) Do there exist perturbations having
less than 6 specials points? b) If the number of special points is 6, may the
number of parabolic curves be smaller than 4? c) If the perturbed surface is
hyperbolic at in nity, may the number of parabolic curves be smaller than
] D.Panov (1997) has constructed perturbed surfaces having one parabolic
curve and 12 special points. B.Segre (1942) has constructed perturbed sur-


Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute


Collections: Mathematics