Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum machine learning due to their desirable geometric inductive biases (e.g., hierarchical structures benefit from hyperbolic geometry). However, going beyond embedding spaces constant sectional curvature, while potentially more representationally powerful, proves be challenging one ca...

The Delaunay triangulation characterizes a natural neighbour relation amongst points distributed in a Euclidean space. In this survey we examine extensions of the Delaunay paradigm that have been used to define triangle meshes for representing smooth surfaces embedded in three dimensional Euclidean space. Progress in this area has stemmed primarily from work done in surface reconstruction and s...

AND Abstract. Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geo-desic ow on any Carnot group is completely integrable. We prove this co...