FMA_40P99_EP - PSC411-MAT (2024-2025)

FMA_41031_EP - Calcul différentiel et fonctions holomorphes (2024-2025)

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.

FMA_41M01_EP - Modal combinatoire, analyse et géométrie (2024-2025)

- 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.

FMA_42032_EP - Distribution, analyse de Fourier et EDP (2024-2025)

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

FMA_42M01_EP - Modal - Algèbre et géométrie - Pavages du plan (2024-2025)

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

FMA_43051_EP - Algèbre et Théorie de Galois (2024-2025)

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

FMA_43052_EP - Analyse fonctionnelle (2024-2025)

This course aims to provide the fundamentals of functional analysis, both as a precursor to applications in (elliptic, parabolic or hyperbolic) partial differential equations and as a precursor to applications in 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: studying convex sets, Helly's Theorem, Hahn-Banach Separation Theorem, and Krein-Milman Theorem.

It then continues with the study of the theorems that form the foundation of functional analysis: Baire's Lemma, Banach-Steinhaus Theorem, Open Mapping Theorem, and Closed Graph Theorem.

Next, we open an important chapter on the study of weak topologies and weak*-topologies, leading us to the statement of the Banach-Alaoglu Theorem (which allows us to "regain" some compactness in infinite-dimensional spaces).

After delving into the study of Banach spaces, we will see to what extent "reflexive" spaces and "separable" spaces constitute an interesting class of Banach spaces that enjoy pleasant properties.

The next chapter is devoted to the study of Banach algebras, which unify several particular cases you may have already encountered (for example, the exponential of a matrix or an endomorphism). This chapter culminates with the proof, in three lines (but requiring understanding of the previous 10 pages!), of a beautiful result concerning Fourier series.

The course continues with the study of the spectrum of operators, particularly Fredholm's alternative, the spectrum of compact operators, and concludes with the study of Fredholm operators, which generalize, in infinite dimension, results that you are familiar with concerning linear mappings between finite-dimensional vector spaces.

Finally, the last chapter of the course is dedicated to unbounded operators and the analysis of operator semigroups, which are the starting point for the study of many evolution equations. In this context, we will prove the Hille-Yosida Theorem (or rather one of its versions known as the Lumer-Phillips Theorem), which provides sufficient conditions for an operator to be the infinitesimal generator of an operator semigroup.

Course language: French
 
Textbook: English