A topological and dynamical approach to the study of complex living systems

Traditionally one has addressed biological phenomena by analytical and/or statistical tools with the goal to give average laws of the competition between biological species and communities and of their evolution. This approach has certainly permitted to obtain important results. However, many fundamental aspects of biological systems at every scale remain to be explained. In the last four decades new mathematical ideas and methods have emerged, relying notably on differential geometry, topology and dynamical systems, which appear to be very meaningful for the modelling of many processes which occur within the cell and organism contexts and activity. In particular, these ideas and methods promises to be helpful for a better understanding of the following three correlated issues (i) the relationship between the plasticity of biological structures and the emergence of their functionality, (ii) the interaction between form and function: how the first influences the second and, conversely, how, the second may favors adaptive changes of the first, (iii) the connection between the robustness and fragility in complex living organism; in particular the difficulty is to characterize the critical thresholds which may occur a transition from robustness to fragility.