(Joint work with P. Pushkar) The four cusp conjecture formulated by V. I. Arnold about 10 years ago is as follows.

Let Vt, t ∈ [0, 1], be a generic smooth family of co-oriented wavefronts in the plane such that V0, V1 are embedded circles with opposite co-orientations, and for each t the front Vt has no oriented self-tangencies. Then there exists t0 ∈ [0, 1] such that Vt0 has at least four cusp points. The proof of this conjecture combines the classical Hurwitz theorem with studying combinatorics of wavefronts.