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.

## A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications

## Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations

.## Geometric evolution problems

Among geometric evolution problems the motion of a surface accordingto its mean curvatureis the best known and it has been widely studied in the last fourdecades. Since singularities mayappear during the evolution, several weak formulations have beenproposed to describe the longtime behaviour of surfaces. One of the possibilities is to representthe initial surface as the level setof an auxiliary (initial) function and then to let evolve all thelevel sets of such a function accordingto the same geometric law.