Recent advances in scanning technology and general progress in science and engineering have made it easy to sample a geometry presented explicitly or implicitly. In some cases, sampling allows the data to be equipped with values of a scalar function defined on the sampled domain. The goal is to process the resulting point cloud data (PCD) for approximating different geometric and topological structures of the original space and/or functions defined on them. How can this be achieved is the main agenda of this award.
The extraction of geometric and topological structures from a PCD begins with the understanding of these structures on the original sampled space. In some cases one needs to adapt existing mathematical concepts to fit the discretization inherent in algorithmic development. In others new analysis techniques and new algorithmic tools need to be developed. PIs expect the more of the latter to tame the curse of dimensionality and scale. The two PIs bring their expertise in computational geometry and computational topology to the board to conduct the project.
A central problem in data analysis is to design computational methods for representing and retrieving information from massive and high-dimensional data. Various scientific, engineering, and social studies such as the ones in product development, medicine, economics, climate, disease control produce data that presumably sample a hidden parameter space sitting in a high dimension. They will benefit from the techniques developed in this project. By the very nature of the problems, this project will enhance a synergy between fields such as mathematics, theoretical computer science, and data analysis.
Other than standard research forums, results from the project will be disseminated through course notes, tutorials, and web-pages to reach wider audience. Graduate students supported by the project will develop skills in mathematics and theoretical computer science, most notably in geometry and topology and also in writing robust, efficient and user-friendly software.
