1ª lição: Around four vertices
The classical 4-vertex theorem asserts that the curvature of a simple closed plane curve has at least four extrema. This theorem, discovered about 100 years ago, has generated a vast literature. I shall discuss this theorem, its numerous ramifications and generalizations, and the theory of evolutes and involutes of plane curves -- a classical, but unfortunately somewhat forgotten, subject.
2ª lição: Web geometry
A 3-web in a planar domain consists of the level curves of three independent functions. The geometry of 3-webs is a fascinating topic which is closely related with the Pappus theorem of projective geometry and the addition of points on a cubic curve. Time permitting, I shall also discuss another type of webs, the Chebyshev nets, that provide a mathematical model for fabric.
3ª lição: Paper sheet geometry
Take a sheet of paper and bend it without folding. The resulting surface has interesting features: it is made of straight lines, and these lines are tangent to a spatial curve. Furthermore, this curve necessarily has cusp singularities. I shall describe the geometry of these developable surfaces and, if time permits, I shall also explain how to fold paper along curved lines.
4ª lição: Segments of equal areas
Consider the family of lines that cut off a fixed area from a given oval in the plane. These lines have an envelope, and this envelope may have cusps. As the value of the area that is cut off from the oval varies, these cusps traverse a curve which, in its turn, has cusps as well. This problem is closely related with the outer billiards, a curious kind of game that is played outside of the billiard table.
5ª lição: The Crofton formula
The object of study of geometrical optics is the set of oriented lines (light rays). In dimension two, this set has an area element invariant under the motions of the plane. The Crofton formula makes it possible to recover the metric in the plane from this area element on the set of rays. Among applications of the Crofton formula one has various geometric inequalities (including the isoperimetric inequality and the Fary-Milnor theorem giving a lower bound on the total curvature of a knot) and a solution to Hilbert's 4th problem: to describe all the metrics whose geodesics are straight lines.