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.

## Advanced FEM Techniques

The Advanced FEM Techniques course is an advanced monographic course on the numerical analysis of finite element techniques. Each year, the state-of-the-art of a research level topic is selected and presented with strong interaction with the students. This year, the course will concentrate on the recently introduced Virtual Element Method, or VEM for short. The VEM is an extremely flexible Finite Element-like mesh based discretisation method allowing for the use of general polygonal/polyhedral meshes.

## Numerical Solution of PDEs

This course provides a high level overview on the numerical solution of partial differential equations. All major classes of numerical methods will be analysed within a rigorous mathematical setting. Key aspects, such as consistency and stability, will be thoroughly investigated, providing the guidelines for the correct choice and implementation of numerical methods for a range of problems.

## Some topics in measure theory

Il corso copre gli argomenti base di teoria geometrica della misura, a partire dall'algebra multilineare fino al teorema di regolarita' locale delle correnti con area minima. Gli argomenti trattati sono:

- alcune nozioni di topologia e distanza di Hausdorff, insiemi analitici
- forme differenziali, algebra esterna
- area formula, coarea formula
- definitione e properita' correnti, compattezza, regolarita'

## 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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