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.
- 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.
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
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
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
This course lays the foundations of functional analysis, both in preparation for its applications to partial differential equations (elliptic, parabolic, or hyperbolic), and as a gateway to the study of operator algebras.
The objective is to provide a broad overview of the theory of Banach spaces and of operators defined between them.
The course begins with geometric considerations: the study of convex sets, Helly’s theorem, the Hahn–Banach separation theorem, and the Krein–Milman theorem.
It then continues with the core theorems that form the backbone of functional analysis: the Baire category lemma, the Banach–Steinhaus theorem, the open mapping theorem, and the closed graph theorem.
We next devote a key chapter to weak and weak* topologies, which naturally lead to the Banach–Alaoglu theorem — a fundamental result that recovers a form of compactness in infinite-dimensional spaces.
After this journey through the “jungle” of infinite-dimensional Banach spaces, we will focus on two particularly important classes: reflexive and separable spaces, which possess especially nice and reassuring structural properties.
The following chapter introduces Banach algebras, which provide a unified framework encompassing various classical examples (such as the exponential of a matrix or of a linear operator). This chapter culminates in an elegant three-line proof — resting on ten pages of theory! — of a beautiful result due to Wiener concerning Fourier series.
We will then study the spectrum of operators, beginning with the Fredholm alternative, followed by the spectral properties of compact operators, and concluding with the theory of Fredholm operators, which generalize classical results from linear algebra to the infinite-dimensional setting. In particular, we will define the Fredholm index, which extends the familiar identity dim Ker u + dim Im u = dim E for any linear map u: E --> F between finite-dimensional vector spaces.
Finally, the last chapter offers an introduction to unbounded operators and the theory of operator semigroups, which provide the natural framework for studying many evolution-type partial differential equations. In this context, we will prove the Hille–Yosida theorem (more precisely, one of its formulations known as the Lumer–Phillips theorem), which gives sufficient conditions for an operator to be the infinitesimal generator of a strongly continuous semigroup.
