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.

## Mechanics of biological systems

- Basic concepts of statistical mechanics, and applications to the mechanics of polymer chains
- DNA structure and basic components of the cell
- Rod elasticity and packing of DNA in viral capsids
- Mechanics of lipid bilayer membranes
- Fluid mechanics: Navier-Stokes equations and low Reynolds number hydrodynamics
- Swimming and locomotion of bacteria and of unicellular organisms
- Motility of crawling cells
- Mechanics of adhesion and of the cytoskeleton

## Introduction to Numerical Analysis

The foundations of Numerical analysis

- Resolution of linear systems with direct methods
- Resolution of linear systems with iterative methods
- Polynomial interpolation and projection
- Numerical Integration
- Numerical solutions of ODEs

Numerical Methods for PDEs

- Finite Elements
- Elliptic Problems
- Parabolic Problems
- Hyperbolic Problems