This research project will carry out development and analysis of several mathematical models. The first set of questions originates in biotechnology. Fast progress in modern biophysics requires new theoretical results on protein solutions (suspensions), particularly in the studies of protein diffusion and of transitions between globular and diffusive states. Diffusion processes in polymer solutions differ from classical diffusion -- while the diffusion coefficient grows linearly with temperature in the classical setting, it decreases near the critical temperature for proteins. A mathematical model for phase transitions in polymers will be developed, allowing one to explain this fact and to explore other near-critical phenomena. One of the possible applications is the analysis of the diffusion of medications in living tissue. The second part of the project concerns population dynamics. Some of the most important questions in this area concern the description of states that are homogeneous in space and time and the study of their stability with respect to local and random perturbations of the environment. These questions will be studied for a wide class of models, including cases with immigration and heavy-tailed migration in the population. The project also includes a study of nonlinear partial differential equations describing the evolution of wave packets in liquids.
The research will be based mostly on the spectral analysis of non-traditional Schrodinger type operators: nonlocal operators and operators on fractals and fractal graphs. A general spectral theory of convolution-type operators and their perturbations will be developed. New results on the inverse scattering problem are anticipated as a result of the last part of the project.
