Good real pictures of complex maps

Abstract: Given a holomorphic map germ f_C : (C^n,0) → (C^(n+1),0), the problem we are interested is f inding a real map germ f_R : (R^n,0) → (R^(n+1),0) such that its complexification is equivalent to f_C and all the topological data of f_C can be found in f_R. More precisely, one wants to find that the equivalent of the Milnor f iber for f_C is realised as a real object.
I will introduce the problem and the (new) techniques we use. After that, I will explain the main ideas to understand our two main results: a restrictive necessary condition to have good real pictures and showing that the inclusion of the real image into the complex is a homotopy equivalence (this was a conjecture from the 90s by David Mond). Time permitting, I will show some counter-intuitive consequences. This is a joint work with Ignacio Breva Ribes.