Energy-driven systems are characterized by two elements: the energy being dissipated and the mechanism by which the energy is dissipated. The dissipation mechanism endows the configuration space with a geometric structure. The investigator studies the geometry of configuration spaces and relevant energy landscapes, and their applications to the dynamics of gradient flows. Gradient flows in Wasserstein metric provide a prime example, given the number of the systems that can be described as such flows and the wealth of results obtained over the last decade. The investigator continues studying gradient flows in the Wasserstein metric, but also examines gradient flows with respect to other, less studied, metrics that appear naturally in physical and biological systems. Understanding the geometry of energy landscapes enables one to make conclusions and predictions about dynamics of complex nonlinear systems. In such systems the detailed dynamics are often practically intractable, but many system-averaged quantities, such as energy, follow precise scaling laws. The investigator studies scaling laws in several energy-driven systems that display coarsening behavior.
The investigator studies complex nonlinear systems relevant to applications in materials science, physics, and population biology. Being able to accurately predict the large-scale features of these systems is important for designing better man-made materials, controlling processes involving phase separation, achieving controlled self-assembly of nano-particles, and understanding (and potentially controlling) the behavior of large-scale animal groups (such as insect swarms). Additionally, he works on geometrical approaches to understanding collections of microscopy images in biology and medicine, in particular, on developing novel tools for image classification and registration, which can lead to a variety of applications to automated processing of biological and medical images.
