Tag
Riemannian Geometry
Riemannian geometry equips a manifold with a metric, and with it lengths, angles, geodesics and curvature. The metric is also what turns a gradient into a vector, since a derivative is naturally a covector and needs a metric before it can be pointed anywhere. Writing here uses Riemannian structure where curvature has physical content, in configuration spaces, in homogenisation, and wherever a shortest path is the object of interest.
Blog
August 9, 2026
Semantic Segmentation as Domain Resolution and Piece Typing
Every segmentation method resolves a domain into pieces and then assigns each piece a type. Reading the field through those two operations puts pixels, point clouds, meshes and shape spaces on one axis, recovers the primal-dual mesh pair on which discrete exterior calculus is built, and gives a precise reason why a message-passing network may place its dual vertices where a discrete Hodge star may not.
April 15, 2026
RVE-to-Mesh Pipeline for Heterogeneous Continua
A single pipeline from microstructure to discrete solver: mean-field homogenisation on a representative volume element produces an SPD permeability tensor field K, the metric (det K)K^-1 turns the Darcy operator into a Laplace-Beltrami operator whose Hodge star discretises the pressure equation, and the scalar curvature of K^-1 drives adaptive refinement.
March 25, 2026
Reservoir Geometry: Riemannian Manifolds in Oil and Gas
Darcy's law recast as geodesic flow on a Riemannian manifold, pressure diffusion as the Laplace-Beltrami equation, and permeability tensor interpolation via SPD geodesics, with no_std Rust for embedded well-site monitoring.