EAPSI: Developing a new theory of discrete angle-preserving surface maps

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The study of complex functions is central to modern mathematics, and may be roughly described as the study of angle-preserving maps between surfaces. A recent application of these maps is their use in the processing of data on two-dimensional surfaces in three dimensions, such as data from medical imaging or 3D facial scans. The maps provide a way to obtain standardized coordinates on such surfaces, to aid data comparison. This project will aim to develop a new finite approximation of these surface maps by proving the Riemann mapping theorem, a foundational theorem in the study of complex functions. This research will be conducted in collaboration with Drs. Jian Sun, David Gu, and Feng Luo, noted experts on computational and low-dimensional geometry and topology, at Tsinghua University in Beijing, China.Given a piecewise flat metric on a compact surface, we may construct it as a gluing of Euclidean triangles, resulting in a geometric triangulation with a finite number of cone points. This research considers a notion of discrete conformality for such metrics that allows for scaling of edge lengths by conformal factors at the cone points, up to Delaunay conditions; and edge flips to transition between Delaunay triangulations. In recent work, a uniformization result was proven, which demonstrated that any such metric is discrete conformal to a metric with equal angle defect (curvature) at each cone point. Furthermore, a discrete Ricci flow realizes this uniformization, and numerous computations of this flow suggest that the resulting piecewise linear maps converge to holomorphic maps as the triangulations become finer. For the case of the disc, attempts will be made to prove this convergence to obtain the classic Riemann mapping theorem. This NSF EAPSI award is funded in collaboration with Chinese Ministry of Science and Technology.