Microlocal analysis

General Informations

This course will be given during the semester 1 of the 2nd year of the Master pathway in Fundamental Mathematics,
which is aimed to prepare students to undertake research in mathematics, either pure or applied.

Course description

Microlocal analysis studies singularities of distributions in phase space by describing their behavior in both position and direction. It has been developed by L. Hörmander, L. Nirenberg and V. Maslov. Two main tools are oscillatory integrals and the method of stationary phase. In this course, we will focus in parallel on:

A) Symbols, Pseudo-differential operators and wavefront sets;

B) Quantum mechanics with a focus on the quantum harmonic oscillator, the uncertainty principle (outlined in the opposite figure), quantization schemes (for Euclidean spaces) and the Stone Von Neumann theorem.

Prerequistes

Students are expected to be familiar with:

Distributions (in particular with the Fourier transform), see for instance the course of F. Golse;

Functional analysis, see for instance the book of W. Rudin;

Spectral theory, see for instance the book by C. Cheverry and N. Raymond.

Historical background

It is preferable to develop some cultural aspects about the quantum aspects
that will be discussed. Students can prepare themselves by looking at:

Le grand roman de la physique quantique from M. Kumar;

Etude de l'équivalence entre mécanique matricielle de Heisenberg
et mécanique ondulatoire de Schrödinger, from L. Coquille.

Educational team

The professor is Christophe Cheverry, see the web page ;
Office 207/1 in building 23 ; contact : e-mail.
The secretariat is provided by Annie Quéméré :
Office 003 in building 23 ; contact : e-mail.

Schedules

The class will be held on Tuesday (from 08h to 10h) and Friday (from 16h15 to 18h15).
It will take place in building 2A, room 329 (tuesday) and in building 2A, room 313 (friday).
It starts the 19/10 and stops the 03/12.

Learning materials

Students will have to refer to their own course notes. We give below some related (recent) references.
Of course, it should be understood that students can also consult the numerous references therein.
To complete the course and to broaden his/her knowledge, students can study the following documents.

- Concerning the part A):

The book about Pseudo-differential Operators and the Nash-Moser Theorem by
P. Gérard (univ. Paris XI) and S. Alinhac is a good classical reference;

The books (volumes I, II, ...) about the Analysis of Linear Partial Differential Operators by
L. Hörmander contain all what is needed;

For a general overview on the analysis of partial differential equations, see the lecture notes by
T. Alazard (ENS). Look especially at the Part 3;

Concerning pseudo-differential operators, look for instance at the introduction to this subject by
J.-M. Bouclet (univ. Toulouse);

- Concerning the part B):

Personal notes about the Stone-Von-Neumann theorem (in french and in english).
Personal notes about the quantum harmonic oscillator (in french and in english).

The quantum harmonic oscillator is discussed (for instance) in the book (Section 6.2) by
C. Cheverry (univ. Rennes 1) and N. Raymond (univ. Angers). Look also at the following
(more physical) discussion and at this introduction of the Segal-Bargmann transform;

The Fourier uncertainty principle is clearly and shortly explained in this text by
S. Fushida-Hardy (univ. Stanford);

Quantization is discussed in-depth (more deeply that we will do) in the lecture notes by
S. Bates (univ. Colombia) and A. Weinstein (univ. Berkeley);

The Stone Von Neumann theorem is presented for instance in the lecture notes by
F. Nier (univ. Paris XIII). This text contains many interesting comments;

The book about quantum theory for mathematicians by
Brian C. Hall (univ. Notre Dame) can give a general overview of the aforementioned material.

Examination

Continuous controls will be carried out regularly. They will take place after
the course of Friday (from 18h15 to 18h45 - 30 minutes).
CC1 the 28/10, with the solutions;
CC2 the 18/11, with the solutions;
CC3 the 25/11, with the solutions;
CC4 the 02/12, with the solutions;
At the end of the course, there will be a two-hour written test.
It is scheduled the 12/12/2022 in the room B02A-302.
Final examination with the solutions.
Annals (2021/22) :
CC1 with the solutions;
CC2 with the solutions;
CC3 with the solutions;
CC4 with the solutions;
CC5 with the solutions.
Final examination with the solutions;