How minimal can be the linear structure of the set of norm attaining functionals of a Banach space?

Ten years ago, the late Charles Read constructed a Banach space which contains no proximinal subspaces of finite codimension greater than one, solving a long standing open problem by Singer of the 1970's. Shortly after that, Martin Rmoutil showed that this space also satisfies that its set of norm attaining functionals contains no two dimensional subspace, solving thus a problem of Godefroy of 2000. There are extensions (and somehow simplifications) of Read's space by Kadets, Lopez, Martin, and Werner. None of these spaces is smooth, so their sets of norm attaining functionals contain nontrivial cones. Even though there are smooth a posteriori versions of Read spaces, their sets of norm attaining functionals also contain nontrivial cones. The aim of this series of talks is to present a detailed account of the very recent construction of a Banach space whose set of norm attaining functionals contains no nontrivial cones (see http://arxiv.org/abs/2406.07273). Actually, the space satisfies that the intersection of any two dimensional subspace of its dual with the set of norm attaining functionals contains, at most, two straight lines (which is, of course, the minimal possibility). We will also show the relation between this example and the long standing open problem of whether finite-rank operators can be always approximated by norm attaining operators. There is no needed background apart of basic knowledge of functional analysis and of the geometry of Banach spaces.