Speaker
Description
Classical methods for geometric data analysis typically rely on constructing appropriate meshes, triangulations or lattices. This process is often computationally expensive, sensitive to noise, and restricted to the “manifold hypothesis”. This talk presents joint work with Iolo Jones that introduces a new, mesh-free framework called diffusion geometry that operates directly on collections of points in $\mathbb{R}^n$.
By reformulating calculus and Riemannian geometry in terms of diffusion processes, we introduce a general recipe for computing standard objects, such as gradients, differential forms, geodesic distances, and sectional curvature, with potential implications for lattice field theory computations and simulations of quantum gravity. Moreover, this data driven approach provides a pathway towards computing parallel transport and invariants such as holonomy, that have previously been out of reach for computational methods.