The previous lecture considered surfaces in 3-dimensional space, but it is useful to triangulate surfaces immersed in higher-dimensional spaces. Spaces with dimensions greater than 3 are naturally encountered when we're interested in dynamic systems and modeling the motion of articulated systems. One example is the study of the energy landscapes of molecules. A molecule is a collection of atoms that move relative to one another (with a limited range of motion). The different positions taken by the molecule when these deformations occur are called conformations of the molecule. Each conformation can be assigned an energy. The result is the energy landscape of the molecule, the understanding of which remains a major challenge in chemistry and biology. The lecture generalizes the results of the previous lecture and deals with the approximation of general geometric objects of any dimension.
This involves introducing concepts of algorithmic topology and developing a theory of geometric sampling beyond the case of surfaces in three-dimensional space. This is done through the notion of distance function, which leads to very general mathematical results. However, their algorithmic implementation is tricky, and the scourge of dimension makes the techniques used in low dimensions too costly in high dimensions (they depend exponentially on the ambient dimension). The lecture shows how the scourge of dimension can be circumvented by restricting the objects of study to varieties of bounded complexity. In particular, we can efficiently reconstruct intrinsically low-dimensional subvarieties from point clouds immersed in very high-dimensional spaces. We can also construct anisotropic meshes and Delaunay triangulations of Riemannian varieties.
