The proof of Arnold's 4 cusp and 4 vertex conjectures

Each convex smooth curve in the plane has at least 4 points where its curvature has a local extremum; if the curve is generic then it has an equidistant with at least 4 cusps. Arnold formulated, using the language of contact topology, two conjectures that generalize these classical results to cooriented fronts in the plane (fronts are curves that are allowed to have certain singularities). These conjectures has been recently proved by the speaker and P. E. Pushkar'. The formulations of these conjectures and the ideas of the proof will be the subject of the talk.