A probabilistic point of view on Hilbert's Sixteenth problem

Hilbert's 16th problem was posed by David Hilbert at the Paris ICM in 1900 and, in its general form, asks for the study of the maximal number and the possible arrangements of the components of a generic real algebraic hypersurface of degree d in real projective space. This is an extremely complicated problem already for the case of plane curves: the possibilities for the arrangement of the components of such curves grow super-exponentially as the degree goes to infinity. (Notice that the same problem over the complex numbers has a simple solution.) An interesting approach is to look at this problem from the probabilistic point of view, by replacing the word "generic" with the world "random". What is the structure of a random plane curve of degree d? And how is it embedded in the real projective plane?

In these lectures I will introduce some tools which can be used for the study of this problem, combining a bit of representation theory, differential topology and asymptotic geometry. These tools are quite general and quantitative in their nature. Rather than presenting a list of results, I will try to suggest a general way of thinking where they can be effectively used.