Resumen:

A central problem in real algebraic geometry is the topological classification of real algebraic varieties. For example, the famous Hilbert's 16th problem asks for the isotopy classification of real algebraic curves in the real projective plane. In the 70's, the problem was revolutionized by Viro's discovery of a beautiful combinatorial method for constructing real algebraic curves with specific topology. In my talk, I want to present the combinatorial part of the construction, its tropical version, and how it can be used to solve problems about smooth and nodal curves. (joint with Ilia Itenberg and Grigory Mikhalkin)