Title: Research projects with Dr Rathinam
Abstract: This talk is intended for graduate students to learn about my research interests and projects. My general research interests are focused on both deterministic and stochastic dynamical systems. This includes modeling and the analysis of models, as well as numerical approximations. For concreteness, I will describe two projects, both motivated by real world problems.
The first project involves being able to predict diffusivity of solutes in an aqueous polymer gel, and is a collaborative project with an experimental chemical engineer. This is a multiscale problem, where the fine structures of the polymer gel are of a length scale of 10nm, whereas we are interested in the solute behavior over length scales of 100 micrometers or more. One PhD student, Preston Donovan, is working on some aspects of this problem. Our first approach has been to regard the polymer molecules as stationary obstacles in a homogeneous medium (water) where a solute molecule diffuses. The diffusion is modeled as an obstructed random walk, where interactions of the solute with polymer molecules is accounted for via altered rates of jumps in the random walk. Recent results obtained by Preston Donovan and I assume a periodic gel, and show that under suitable conditions, the large scale behavior of the solute may be approximated by Brownian motion with certain diffusivity matrix. This matrix can be computed by solving a large linear algebra problem based on the details of the finescale model. We describe both, a formal asymptotic expansion and a stochastic process limit theorem. I will describe some unsolved problems and questions in this regard. Some other aspects of this project include developing a better understanding of polymer gel physics, developing better statistical inference of the random walk from experimental measurements.
The second project involves modeling and analysis of behavior of social groups. The particular aspect of focus here is what happens collectively to a group when individuals influence and are influenced by each other according to certain basic rules of interaction. This includes modeling the opinions of people in a group as well as swarming behavior of birds and insects. A recent UMBC PhD graduate Dr Serap Tay Stamoulas, worked on the analysis of a multidimensional deterministic opinion dynamics model as well as a scalar binary (yes or no) stochastic opinion dynamics model. I will describe some future questions I wish to explore in this area.