## Advanced Programming

The course aims to provide advanced knowledge of both theoretical and practical programming in C++14 and Python3, particularly the principles of object-oriented programming and best practices of software development.

**Syllabus:**

## Introduction to Elliptic Equations

*1. Laplace equation:*

- harmonic functions, mean value properties,
- maximum principle,
- Green's function,
- Poisson kernel,
- Harnack inequality,
- subharmonic functions,
- Perron-Wiener-Brelot method for the Dirichlet problem,
- regular boundary points.

*2. Variational theory of elliptic equations:*

## Advanced Analysis - A

0. Metric and normed spaces, converging and Cauchy sequences, completeness, Banach spaces. Completion theorem. Dimension, finite-dimensional spaces and infinite-dimensional spaces. Space of real polynomials on an interval; space of real continuous functions on an interval. Compactness, sequential compactness, precompactness, relative compactness and their relations. Density and separability. Linear bounded operators on Banach spaces and their basic properties. Extension theorem.

## Advanced Mathematical Physics A

See homepage of Trieste University

## Advanced Mathematical Physics B

See homepage of Trieste university

## 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.