Contrary to what some people can say, non-singular plane curves are not boring! If the classification of complex curves is not very difficult, that of their real counterparts is so complicated that Hilbert's 16th problem still stands to this day, 124 years after its formulation. We are currently stuck in degree 8.
I will review the topological approach, and describe how one could try to answer the question posed by Hilbert: restrictions and constructions. In fact, this talk will be a way for me to tease algebraic geometers, since I will try as much as possible to forget about algebraic things. Whoopsie!
