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.

## Interacting Diffusions, Optimal Transport and Extended Metric Measure Geometry

Abstract: The study of interacting particles has been a central subject in statistical physics. The goal of this course is to study interacting infinitely many Brownian motions with long range interaction by using methods from extended metric measure geometry and optimal transport theory. The lectures will cover several topics from the following subjects:

## Geometry and isoperimetry on manifolds with Ricci lower bounds

Isoperimetry here is intended in a broad sense, comprising the study of isoperimetric problems on Riemannian manifolds in connection with the validity of functional, isoperimetric and spectral inequalities.

## An optimisation–based domain–decomposition reduced order model for parameter–dependent non–stationary fluid dynamics problems

## Introductory lectures on mean curvature flow and minimal surfaces

**Lesson 21/05: ROOM 134****Lesson 22/05: ROOM 133****Lesson 28/05: ROOM****Lesson 29/05: ROOM 136**

Program: