*Note from June 29, 2020*

Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.

5 credits

30.0 h

Q1

Teacher(s)

Ringeval Christophe;

Language

English

Prerequisites

Having followed LPHYS1202 is an asset.

Main themes

This teaching unit aims at presenting and deepening the mathematical structures supporting the construction of modern physics theories. These structures will be presented according to the logical flow in which they can be constructed. Various practical examples taken from actual physics will be used as an illustration of their importance.

Aims

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a. Contribution of the teaching unit to the learning outcomes of the programme (PHYS2M and PHYS2M1)1.2, 2.1, 2.5, 3.1, 3.2, 3.3, 3.4 b. Specific learning outcomes of the teaching unit At the end of this teaching unit, the student will be able to : 1. express the axioms supporting the mathematical structures seen in the lectures ; 2. express and demonstrate the main theorems used in physics ; 3. generalize and apply the techniques seen in the lectures to new problem in physics. |

*The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.*

Content

The lectures follow the following tree :

- Concepts of topology

* Euclidian

* Connected space, topological group

- Measure theory and Lebesgue integral

* Measurable space and functions

* Lebesgue's integral

* Applications to probabilities

- Distributions et Green's functions

* Tests functions and distributions

* Operations and Fourier transforms

* Green's functions

- Spectral theory in Hilbert's spaces

* Elementary properties of Hilbert's spaces

* Linear functional and operators

* Spectra of bounded operators

* Unbounded operators, self-adjoint, symmetric

* Spectral theorem

- Concepts of differential geometry

* Manifolds and differential forms

* Flow, Lie derivatives and commutators

* Exterior derivative

- Concepts of topology

* Euclidian

* Connected space, topological group

- Measure theory and Lebesgue integral

* Measurable space and functions

* Lebesgue's integral

* Applications to probabilities

- Distributions et Green's functions

* Tests functions and distributions

* Operations and Fourier transforms

* Green's functions

- Spectral theory in Hilbert's spaces

* Elementary properties of Hilbert's spaces

* Linear functional and operators

* Spectra of bounded operators

* Unbounded operators, self-adjoint, symmetric

* Spectral theorem

- Concepts of differential geometry

* Manifolds and differential forms

* Flow, Lie derivatives and commutators

* Exterior derivative

Teaching methods

The teaching methods is traditional lecturing on the black board alternated with inquiry-based methods during collective discussions.

Evaluation methods

Evaluation is performed with a 2-hours long written exam dealing with the subjects and methods addressed during the lectures, but also with their application to new problems which have not been explicitly solved in the course.

Bibliography

- Geometry, Topology and Physics, Nakahara.

- Méthodes mathématiques pour les sciences physiques, Schwartz.

- Lebesgue Measure and Integral, Craven.

- Méthodes mathématiques pour les sciences physiques, Schwartz.

- Lebesgue Measure and Integral, Craven.

Faculty or entity

**PHYS**