Quasiconformal analysis, optimal triangulations and fractal geometry

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The project will use ideas from classical real and complex analysis to solve problems arising in computer science, physics, dynamics and probability. One focus of the project is optimal meshing and triangulation, which has many applications, from computer graphics to modeling fluid flows. In many problems it is necessary to approximate a region of space as a finite union of triangles. The shapes of the triangles greatly influences the accuracy and speed of various numerical methods, and it is important to avoid angles that are too small or too large. Finding good triangulations is often the most time consuming (and least automated) step in numerical modeling. Funding for this project will lead to the development of algorithms that compute the best possible angle bounds for a given domain and find practical triangulations achieving these bounds. Other parts of the funded work will develop methods to recognize and classify 2-dimension shapes via special embeddings into 3-dimensional space, and will give improved estimates for certain random growth models that have been used to model tumors and the propagation of fluids through porous materials. The project provides research training opportunities for graduate students. This work will extend the conformal and quasiconformal methods introduced by the PI for triangulation of polygonal planar domains to more difficult situations involving PSLGs (planar straight line graphs), non-planar surfaces, and 3-dimensional regions. The PI's previous work involves dynamical flows associated to triangulations and discrete analogs of closing lemmas for these flows. These results will be extended and generalized. The work will also expand on the PI's results on the smoothness and geometry of Weil-Petersson curves, aiding computation with these curves, which are closely related to string theory, pattern recognition, geometric measure theory and random curve families. In this project, planar curves are associated to minimal surfaces in hyperbolic 3-space in such a way that geometric properties of the curve are reflected in the properties of the surface. The PI will also investigate growth estimates for diffusion limited aggregation, a random growth process from statistical physics that is motivated by a number of natural processes, but about which few rigorous results are known. 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.