Due to rapid and vast data acquisition by modern technology, experiments and computer simulations, computational and data analytical approaches have become increasingly important for many important applications in science and engineering. An important but challenging task in data representation, analysis and understanding is to extract information of interest, which includes intrinsic geometric features and global structures from large data set. The aim of this project is to introduce advanced mathematical theories and analytical tools in differential geometry and translate them into efficient computation algorithms for data analysis.
The main effort of this project is to develop new mathematical models and computational tools for a few important but challenging tasks in 3D modeling and data analysis in higher dimensions. These models and tools will be utilized to extract and characterize geometric structures for data analysis in various applications. In particular, quasi-conformal map and conformal structure will be used for representation and analysis for surface maps and shape modeling. Intrinsic geometric differential operators, such as Laplace-Beltrami operator and its eigen-system, will be explored for point cloud analysis and manifold learning in high dimensions. Models, methods and computational tools developed in this project will be tested on benchmark data sets as well as real applications. Interdisciplinary applications and collaborations with computer scientists and statisticians will also be pursued.
