Talk time: 10:00-10:50 am

Name: André Belotto da Silva, Université Aix-Marseille, France

andre-ricardo.BELOTTO-DA-SILVA@univ-amu.fr

This talk concerns Gabrielov's Rank Theorem, a fundamental result in local complex and real analytic geometry, proved in the 1970's. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is behind several pathological examples in local real analytic geometry. Gabrielov's Rank Theorem provides a formal condition for the equality to hold.

In spite of its importance, the original proof is considered very difficult. There is no alternative in the literature, besides work by Tougeron, which is itself considered very difficult. I will present our recent work in collaboration with Octave Curmi and Guillaume Rond, where we provide a proof of Gabrielov's Rank Theorem. Indeed, we develop formal geometric techniques, inspired by ideas of Gabrielov's and Tougeron's, in order to clarify the difficult part of the proof.