This course is a basic mathematics course providing the tools used in applied mathematics, physics, mechanics and economics.

It also prepares students for more advanced mathematics courses, in particular those of the M1 program.

The first part (5 blocks) is devoted to the theory of holomorphic functions and the second part (5 blocks) to differential calculus.

- Teaching coordinator: Deroin Bertrand
- Teaching coordinator: Han-Kwan Daniel
- Teaching coordinator: Leguil Martin
- Teaching coordinator: Maleze Cyril
- Teaching coordinator: Renard David

- Teaching coordinator: Pichon Eric
- Teaching coordinator: Renard David

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

- Teaching coordinator: Golse François
- Teaching coordinator: Jendrej Jacek

- Teaching coordinator: Alves Sampaio Paulo Elpidio
- Teaching coordinator: Golse François

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

- Teaching coordinator: Bijakowski Stéphane
- Teaching coordinator: Fantini Lorenzo
- Teaching coordinator: Ngo Dac Tuan

- Teaching coordinator: Bijakowski Stéphane

This course intends to provide the foundations of Functional Analysis having in mind applications to partial differential equations and applications to operator algebras.

The objective of the course is to give a fairly general overview of the study of Banach spaces and operators between Banach spaces.

The course begins with geometric considerations: study of convex sets, Helly's Theorem, Hahn-Banach's convex separation Theorem, Krein-Milman's Theorem.

Then, it continues with the study of theorems which form the basis of functional analysis: Baire's Lemma, Banach-Steinhaus Theorem, Open Mapping Theorem and the Closed Graph Theorem.

We then open an important chapter on the study of weak topologies and weak-∗ topologies, which will lead us to the statement of the Banach-Alaoglu Theorem (which allows us to “recover" some compactness in infinite dimensional spaces ).

After getting a little lost in the study of very general Banach spaces, we will see to what extent “reflexive” spaces and “separable” spaces constitute an interesting class of Banach spaces, since they enjoy pleasant properties.

The next chapter is devoted to the study of Banach algebras which unify under the same banner several special cases that you may have already seen (e.g. exponential of a matrix or a linear map). This chapter culminates with the proof in three lines (but which requires having understood the previous 10 pages!) of a beautiful result on Fourier series.

The course ends with the study of the spectrum of operators with in particular the Fredholm alternative, the study of the spectrum of compact operators to culminate with the study of Fredholm operators, which generalize in infinite dimension, the results that you know well of course linear maps between finite-dimensional vector spaces.

**Course language:**French

**Lecture notes:**English

- Teaching coordinator: Laurent Camille
- Teaching coordinator: Morabito Francesco
- Teaching coordinator: Pacard Frank
- Teaching coordinator: Prange Christophe

- Teaching coordinator: Morabito Francesco
- Teaching coordinator: Pacard Frank

- 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.

- Teaching coordinator: Amini Omid

In this modal, we'll explore the notion of tessellation(or tiling), and through this, that of groups and actions of groups. We'll tackle Bieberbach's classic results on regular tessellations of the plane, Penrose's famous aperiodic tessellations, and affine tessellations of the plane.

**References:**

Tessellations of the plane, notes from a mini-course given at the École Polytechnique

http://www.math.polytechnique.fr/xups/xups01.01.pdf

**Course language:** French

- Teaching coordinator: Tessera Romain