Abstract : Emergence is a process by which coherent structures arise through interactions among elementary entities without being directly encoded in these interactions. In this course, we will address some of the key questions of emergence such as the deciphering of the hidden relation between individual behavior and emergent structures. We will start with presenting biologically relevant examples of microscopic individual-based models (IBM). Then, we will develop a systematic coarse-graining approach and derive corresponding coarse-grained models (CGM) using mathematical kinetic theory as the key methodology. We will highlight that novel kinetic theory concepts need to be developed as new mathematical problems arise with emergent systems such as the lack of conservations, the build-up of correlations, or the presence of phase transitions (or bifurcations). Our goal is to show how kinetic theory can be used to provide better understanding of emergence phenomena taking place in a wide variety of biological contexts.

Keywords : packing; non-overlapping constraint; minimization; level sets; continuity equation; domain velocity; Dirichlet energy; Laplace-Beltrami