The PI studies two lines of applications of systems with nonlocal effects, that is systems in which points are subject to far-away influences. One application is to investigation of large data sets. Advances in automated data-acquisition techniques (such as digital microscopy) has enabled one to gather large data sets holding a wealth of information. Such data sets are often of very high dimension (for example digital images). To be able to extract the information contained, it is desirable to simplify them with minimal loss of information. One approach is to approximate the data set by a low-dimensional manifold. This project is concerned with a variational approach using nonlocal energies for parameterizing the data with a curve. The goal is to contribute to understanding of which functionals provide for good data approximation and are amenable to efficient computational implementation. The PI studies the fundamental questions such as existence and regularity of minimizers, as well as numerical implementation and applicability to data sets. The other line of investigation concerns behavior of particle systems with long range interaction. Such systems arise as models of collective behavior of a variety of living organisms (locust, fish, birds, bacteria, and others). Goal of this research is to help explain why and how large, well-organized groups (such as swarms, schools, flocks, etc.) of organisms form. Furthermore to describe mathematically the evolution of these groups, and explain why are they stable and explore how they interact with the environment in particular in the presence of environmental boundaries. The development of the new mathematical tools, such as gradient flows in spaces of probability measures, provides the techniques that make significant progress on these issues likely.
Being able to extract information from large data sets is a scientific challenge with important practical consequences. The research of the PI can lead to improved algorithms for data parameterization and approximation. This in turn can improve data clustering, classification, visualization, and other tasks. An example of an application is improving and automating the diagnostics of some diseases, based on images of tissue samples. Understanding how large groups of a variety of organisms form and behave is an important biological question. It is also one where mathematics, based on simple rules of interaction between individuals, can provide important insights. The knowledge obtained can be used to predict and offer guidance on how such groups (for example locust swarms) could be influenced.
