The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

## Calculus of variations

Programme: Minumum problems for integral functionals in one independent variable: necessary and sufficient optimality conditions, solution to classical problems, problems invariant under reparametrization and geodesics. Minumum problems for multiple integrals: direct methods, lower semicontinuity and relaxation results for multiple integrals, quasiconvexity and polyconvexity.

Number of lectures: 30 two hour lectures

Period: November 15 - March 29