# Arnold's Problems by Vladimir Igorevich Arnol'd Vladimir Arnold is likely one of the most eminent mathematicians of our time a lot of those difficulties are on the entrance line of present research.

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Is it true that in the complex case the complement of the bifurcation diagram of a function is always a K(%, 1) space? Are the components contractible in the real case? Conjecturally no, although R. Thorn had thought that yes! 1975-12. Does every real-valued function have a real Modification (with \i real critical points)? 1975-13. What is the minimal number of critical values obtained by a perturbation of a critical point of multiplicity \i with (i Morse critical points? Conjecturally it is n + 1, where n is the number of variables (or corank).

Investigate the bifurcations of type D5 in the 3-space topologically {the problem has been studied by V. I. Bakhtin). 1978-20 The Problems 37 1978-20. Investigate the singularities of bicaustics of type D\$, up to diffeomorphisms. 1978-21. Investigate the process of sweeping the bicaustic D\, up to equivalence (strong equivalence): given three smooth curves (p,-: (M, 0) —>• (M2,0) starting from 0 with the same velocity v ^ 0, in all other generic. The equivalence is provided by the diagrams (R,0) - ^ ^ (R 2 ,0) (where the diffeomorphisms x and h are independent of /); the strong equivalence is: x(t) = t + const.

1980-6. ). 1980-7. Construct a theory of caustic cobordisms (different from that of Lagrangian cobordisms). 1980-8. In the theory of singularities (e. , critical points of functions), why is the codimension in the real case the same as in the complex case? Compare with the R- and C-modality and with the {co)dimension of the prolonged self-intersection line of the swallowtail or the umbrella. 1980-9. Apply mixed Hodge structures to real algebraic geometry. For example, for estimation of topological invariants of real Morsifications, and for investigation of the topology of discriminants.