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.

## Functional analysis

Aim of the course is to introduce the basic tools of linear and nonlinear functional analysis, and to apply these techniques to problems in PDEs. The course is divided into two parts: the first one concerns spectral theory of linear operators, whose goal is to extend the classical notion of spectrum of a matrix to an infinite dimensional setting. The second part of the course introduces the methods of nonlinear analysis to find the zeros of a nonlinear functional on a Banach space. In particular it gravitates around the implicit function theorem and its variants.

## Topics in Continuum Mechanics

- Reminders on linear algebra and tensor calculus
- Kinematics of deformable bodies
- Eulerian and Lagrangian descriptions of motion
- Balance laws of continuum mechanics: conservation of mass, balance of linear and angular momentum, energy balance and dissipation inequality
- Constitutive equations
- Fluid dynamics: the Navier Stokes equations
- Solid mechanics: nonlinear and linearized elasticity
- Selected topics from the mechanics of biological systems

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