This proposal pretends to interlace tools from singularity theory and differential geometry in thestudy of frontals in high co-dimension, mainly focused in minimal surfaces with singular points and branched immersions in high dimensions, generalizing tools of my thesis.In my doctoral thesis, we introduce tools to investigate the differential geometry of singular surfaces known as frontals, in the Euclidean space. These tools allow us to explore the geometric behavior close to the singularities of the most degenerate types. It is possible to prove that the singularities that occur on minimal surfaces are frontals and degenerate, so our methods apply. We want to generalize the tools and concepts in to higher dimensions in order to apply these to minimal surfaces in higher dimensions. The importance of these minimal surfaces with singularities, emerged in the solutions to the Plateau's problem and other geometrical problems of calculus of variations. For the minimizing solutions of Plateau's problem in the space $\R^3$, Robert Osserman proved that they do not have singularities, but in the case of $\R^n$ with $n\geq4$ there are examples of minimizing solutions with singularities of maximum co-rank, from this the importance of studying them.We also intend to study curves, tangent cones and the Nash fiber on the direct image of an analytic or definable frontal in an o-minimal structure.
News published in Agência FAPESP Newsletter about the scholarship: