MATH-405 / 5 credits

Teacher:

Language: English

Remark: pas donné en 2024-25. Cours donné en alternance tous les deux ans

## Summary

An introduction to methods of harmonic analysis. Covers convergence of Fourier series, Hilbert transform, Calderon-Zygmund theory, Fourier restriction, and applications to PDE.

## Content

-Fourier series, convergence and summability.

-Fourier series, convergence and summability.

-Hilbert transform.

-Calderon-Zygmund theory of singular integrals.

-Liitlewood-Paley theory.

-Fourier restriction.

-Applications to dispersive PDE.

## Keywords

Fourier series, convergence, singular integrals, Calderon-Zygmund theory, Fourier restriction.

## Required courses

Analyse I - IV, Algebre lineaire I et II.

## Recommended courses

Analyse I - IV, Algebre lineaire I et II.

## Important concepts to start the course

Understand key concepts of real analysis, such as measure and Lebesgue integral. Be able to construct a rigorous mathematical argument.

## Learning Outcomes

By the end of the course, the student must be able to:

• Analyze convergence of Fourier series
• Examine bounds for singular integrals
• Prove bounds for dispersive PDE

## Transversal skills

• Communicate effectively with professionals from other disciplines.
• Access and evaluate appropriate sources of information.
• Give feedback (critique) in an appropriate fashion.

## Teaching methods

Two hours ex cathedra lectures, two hours of exercises led by teaching assistant.

## Expected student activities

Attend lectures and exercise sessions, read course materials, solve exercises.

## Assessment methods

Oral exam at the end of course.

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de '¿examen qu'il communique aux étudiants concernés.

## Supervision

 Office hours No Assistants Yes Forum No

## Bibliography

-Classical multilinear harmonic analysis by C. Muscalu and W. Schlag.

-Singular integrals and differentiability properties of functions by E. Stein.

No.

## In the programs

• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Harmonic analysis
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Harmonic analysis
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Harmonic analysis
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

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