Ingénieur 2A

This course provides basic training in analysis. This module enables students to master the mathematical tools used in applied mathematics, physics, mechanics and economics. It paves the way for third-year advanced mathematics programs.

The course introduces the formalism of distributions, introduced by Laurent Schwartz in the late 1940s, which provides a natural framework for the study of Fourier transformation. It then focuses on the  study of the fundamental properties of the main partial differential equations of mathematical physics

- Distributions, derivation, convolution, regularization.
- Fourier series and transformations.
- Poisson and Laplace equations. Harmonic functions.
- Heat equation.
- Wave and Schrödinger equations.

F. Golse: "Distributions, analyse de Fourier et équations aux dérivées partielles"

Appendix "Intégration sur les surfaces"

Course language : French

Galois theory emerged in 19th century to study the existence of formulas for solutions of polynomial equation (in terms of the coefficients of the equation). The theory is both powerful and elegant and was the origin of a very large part of modern algebra. Nowadays it is also a very active research field.

The aim of this course is first to introduce basics and tools of general algebra (groups, rings, algebras, quotients, field extensions...) which will allow in the second part of the course to develop Galois theory, as well as some of its most remarkable applications.

Beyond the the interest on the subject for itself, the course aims at being a good introduction to algebra and its applications, in Mathematics and in other fields (for instance Computer science with finite fields, Physics and Chemistry with group theory).

*Prerequisites

Standard linear algebra from the first two years at University.

* Knowledge expected at the end of the course : 

Theoretical knowledge :

- Knowledge of fundamental structures in general algebra.

- Knowledge of fundamental concepts in Galois theory (Galois extensions, Galois group)

- Most important examples (finite fields, cyclotomic extensions solvable extensions).

- Main historical applications (solvable polynomial equations, constructability of regular polygons).

Practical knowledge :

- Handling of fundamental algebraic structures, computation of degrees of extensions.

- Characterization of Galois extensions.

- Computation of Galois groups, method of reduction modulo p.

- Applications of the theory, in particular to number theory and fields theory

* Evaluation : exam at the end of the course.and one homework

Language  : French

- Hodge theory and exotic geometries

- Since Poincaré, the mathematical properties of topological spaces have been studied by associating algebraic invariants with them.

When the topological space is provided with a suitable metric, it is possible to use analysis to represent these algebraic invariants as solutions of a Laplace equation on the geometric space in question. This is the subject of a mathematical theory named after the Scottish mathematician William Hodge.

What happens when the geometric structure degenerates and space becomes singular? A central question with multiple links to physics by its nature.