The principal investigator and his colleagues develop computational and theoretical framework to analyze large data sets with low-dimensional intrinsic structure. More specifically, they address the following challenges: Constructions of underlying curves and surfaces in the presence of significant outliers and noise; Improvement of recent nonlinear embedding techniques for large data sets with significant noise; Analysis of large data sets generated by special nonlinear partial differential equations with low-dimensional inertial manifold. There are several important applications of the proposed research: quantitative edge detection in images, detection of nuclear devices by muon radiation, identification of protein-binding genomic regions (and even specific sites), quantitative exploration of the functional domain in the gene ontology and its relation with structural properties.
The broader impacts of the proposal are as follows: 1) The mathematics suggests important applications, some of them are listed above. 2) The applications guide and demand a broad framework for multiscale geometric analysis of data sets with intrinsic low-dimensional geometric structures. 3) Interaction between different areas of mathematics, in particular, computational harmonic analysis, scientific computation, statistical learning, probability and mathematical modeling. 4) Multidisciplinary collaborations, involving applied mathematicians, biologists, computer scientists, statisticians and mathematical analysts. 5) Industrial collaborations. 6) Training of young researchers in a promising new area of mathematics.
