Lins between microscopic and macroscopic descriptions

The problem of relationships between various scales of description seems to be one of the most important problems of the mathematical modeling of complex systems (including e.g. the modeling of solid tumor growth). In the lecture we provide the theoretical framework for modeling at the microscopic scale in such a way that the corresponding models at the macro-, meso- and micro-scales are asymptotically equivalent, i.e. the solutions are close to each other in a properly chosen norm. In mathematical terms we state the rigorous links between the following mathematical structures: 1. Continuous (linear) semigroups of Markov operators: the micro-scale of stochastically interacting entities (cells, individuals,...); 2. Continuous nonlinear sernigroups related to the solutions of bilinear Boltzmann-type nonlocal kinetic equations: the meso-scale of statistical entities; 3. Dynamical systems related to bilinear reaction-diffusion-chemotaxis equations: the macro-scale of densities of interacting entities.