Mathematical biology offers a rich set of topics, challenges, and interesting questions. It can motivate new mathematical problems and contribute to a better understanding of complicated biological systems, including those in cell biology. In this lecture, I will describe some of the research carried out in my group over the years, with contributions from many talented colleagues and trainees. I will focus mainly on relationships between local (PDE) and nonlocal variants of models that describe the internal structure and the interactions of cells. I will also survey work on reaction-diffusion systems associated with cell motility and recent experimental validation of some of our models. ?
Discs are among the simplest manifolds, but their groups of diffeomorphisms can be very complicated. I will describe the techniques from geometry, topology, and dynamics that were used to understand these groups in low dimensions, the relationship of these groups to stable homotopy theory and number theory in high dimensions, and recent breakthroughs in understanding their rational homotopy type. This talk will be aimed at a broad audience.