## Introduction to numerical analysis and scientific computing with python

Syllabus 2023-2024

- Basics on Scientific Computing
- Vector spaces, vector norms, matrices, and matrix norms
- Basic linear algebra: direct solution of linear systems
- Not so basic linear algebra: iterative solution of linear systems
- Polynomial interpolation
- Interpolatory Quadrature rules
- L2 projection / Least square approximation
- Introduction to Finite Difference Methods
- Introduction to Finite Element Methods

Python laboratories

## Algebraic geometry

The course aims to introduce the student to the language of *schemes*, the central object of study in modern algebraic geometry.At the end of the course the student will master the basic dictionary, shared by all algebraic geometers,regarding the theory of schemes, affine and projective varieties, their morphisms and cohomology of coherent sheaves on them.All fundamental results will be presented with fully detailed proofs, and exercises will be proposed constantly during the course.

## Linear and nonlinear bifurcation problems (Topics in Ad. Analysis 2)

After introducing the theory of analytic functions between Banach spaces, we shall present perturbative results for the spectrum of linear operators, in particular for separated eigenvalues of closed operators, with applications to the stability of traveling water waves. Then we shall present bifurcation results of periodic and quasi-periodic solutions of nonlinear dynamical systems as well as homoclinic solutions to hyperbolic equilibria of Hamiltonian systems.

## Introduction to analytic number theory (Topics in Ad. Analysis 1)

This course considers the classical topics in analytic number theory, with a focus on tools coming from Fourier analysis. The list of topics to be covered is as follows:

1. Review of the basic elements of Fourier analysis: Fourier transform in L^1 and L^2; Plancherel's theorem; Tempered distributions; Fourier series; Convolution and approximations of the identity.

2. Diophantine approximations; Equidistribution of sequences; Notions of discrepancy; Erdös-Turán inequality; Irregularities of distribution.

## Advanced analysis - A

Program of the course Advanced Analysis –A (2023-2024)

## Advanced Geometry II

Smooth manifolds and differential topology

## Reduced Order Methods for Computational Mechanics

**Mathematics Area, PhD in Mathematical Analysis, Modelling and Applications (AMMA)**

**Master in High Performance Computing (MHPC)**

**Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Michele Girfoglio, Dr Niccolò Tonicello and Nicola Demo.**