Mathematics shapes our world: it incessantly develops, asking for solutions and ideas. At the same time, its beauty amazes us and, surprisingly, even the most abstract theory may apply to practical purposes. The research activity in Mathematics at FIM embodies both these aspects, as it covers the theoretical side as well as the applied approach, proving the matter to be far from cold, but lively and challenging. The results of our efforts are an interacting community whose specific fields of interest are: