These lecture notes address mathematical issues related to the modeling of impact laws for systems of rigid spheres and their macroscopic counterpart. We analyze the so-called Moreau's approach to define multibody impact laws at the mircroscopic level, and we analyze the formal macroscopic extensions of these laws, where the non-overlapping constraint is replaced by a barrier-type constraint on the local density. We detail the formal analogies between the two settings, and also their deep discrepancies, detailing how the macroscopic impact laws, natural ingredient in the so-called Pressureless Euler Equations with a Maximal Density Constraint, are in some way irrelevant to describe the global motion of a collection of inertial hard spheres. We propose some preliminary steps in the direction of designing macroscopic impact models more respectful of the underlying microscopic structure, in particular we establish micro-macro convergence results under strong assumptions on the microscopic structure.

<p style='text-indent:20px;'>We study the extension of the macroscopic crowd motion model with congestion to a population divided into two types. As the set of pairs of density whose sum is bounded is not geodesically convex in the product of Wasserstein spaces, the generic splitting scheme may be ill-posed. We thus analyze precisely the projection operator on the set of admissible densities, and link it to the projection on the set of measures of bounded density in the mono-type case. We then derive a numerical scheme to adapt the one-typed population splitting scheme.</p>