New trends in combinatorial patchworking

(Joint with Erwan Brugallé and Lucía López de Medrano.) Combinatorial patchworking is a technique introduced by Viro to study the topology of real algebraic hypersurfaces. It's base ingredients are a triangulation of a lattice polytope and a sign distribution on its vertices. If the triangulation is "convex", the construction can be translated to the dual setting of tropical varieties. In recent years, this dual viewpoint has promoted new results using tropical homology theories. For example, Renaudineau and Shaw proved a conjecture by Itenberg bounding the Betti numbers of the real part of a (unimodular) patchwork hypersurface in terms of the Hodge numbers of the complex part. In our work, we go back to the original setting and show that many of the new results hold in the non-convex case and, to some extent, in higher codimensions. In my talk, I will try to give a brief overview of these results and their “new” formulation in the “old” setting.