Discrete Conformal Geometry of Surfaces and Applications

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Digital surfaces are being massively produced from scanners, imaging systems, sensors and many other devices these days. How to compare and categorize these surfaces is an urgent and important problem. This National Science Foundation funded project aims to develop mathematical theories for an efficient and speedy classification of digital surfaces. A successful completion of this National Science Foundation funded project will have applications in medical imaging, game industry, engineering and many fields. Some of the earlier work of the PI in this area have already been used in health and game industries. The project work will further advance this progress and develop deep mathematics based on the classical Riemann surface theory and conformal geometry. The classical theory of Riemann surfaces is a powerful tool for classifying surfaces up to conformal diffeomorphisms. The most important theorem in the theory is the Poincare-Koebe's uniformization theorem. The theorem has a wide range of applications within and outside mathematics. However, computation of uniformization maps and metrics on non-flat surfaces are very difficult in general. The goal of the proposal is to develop a theory of discrete conformal geometry for polyhedral surfaces, to establish the counterpart of the uniformization theorem in the discrete setting, and to prove convergence of discrete conformal maps to conformal maps. The PI will develop efficient algorithms to compute uniformization metrics through collaboration. The main ingredient in developing a discrete conformal geometry is to define the notion of discrete conformality. In a recent joint work with collaborators, the PI introduced a notion of discrete conformal equivalence for polyhedral metrics on surfaces. The PI and his collaborators proved a discrete version of uniformization theorem for compact polyhedral surfaces and showed that discrete conformal maps converge to conformal maps on tori and disks. There remain several major open problems of establishing the convergence of discrete conformal maps for all surfaces and proving the discrete uniformization theorem for all non-compact simply connected surfaces. The discrete conformality introduced by us is closely related to the work of Alexandrov, Pogorelov and Thurston on convex surfaces in hyperbolic three-space, the classical Wyle problem and the Koebe circle domain conjecture. Tools developed in the classical area of mathematics will be used to solve the proposed problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.