Enumerations of real curves

Ponente(s): Erwan Brugallé Bouttier

Enumerative geometry is the area of mathematics which studies questions like: how many lines pass through two points (easy)? How many conics pass through five points (easy)? How many cubics with a crossing point pass through 8 points (less easy)? When dealing with algebraic curves defined over the field of complex numbers, this number of curves does not depend on the chosen configuration of points, in the same way that the number of roots of a complex polynomial in one variable does not depend on its coefficients but only on its degree. However, when dealing with algebraic curves defined over the field of real numbers, this number heavily depends on the chosen configurations of points, and the problem becomes much more intricate. Recent years have seen a tremendous development in enumeration of real curves, mainly based on the work by J.-Y. Welschinger. In this talk, I will give an introduction to (real) enumerative geometry and I will present some recent development in the field. I will in particular make a detour via tropical geometry and tropical refined invariants.