College of Liberal Arts and Sciences

# Department of Mathematics

## Course Info

### MATH 5141: Abstract Harmonic Analysis - Fourier Analysis

Description: Harmonic analysis on various spaces such as Euclidean spaces, and abelian and non-abelian locally compact groups. We will use the book A Course in Abstract Harmonic Analysis by Gerald Folland (http://www.amazon.com/Abstract-Harmonic-Analysis-Advanced-Mathematics/dp/0849384907) as the textbook. This is an introduction to Fourier analysis on groups and its connections to unitary representation theory. Some of the topics to be covered include locally compact groups, Haar measure, and unitary representations, going into more details on Fourier analysis on abelian groups (dual groups and Pontryagin duality) and compact groups (the Peter-Weyl Theorem).

Offered: Spring

Credits: 3

### MATH 5141 - Section 1: Abstract Harmonic Analysis

Description: The goal of the course is to cover several modern topics in Harmonic Analysis. In particular we are going to cover the following material:

1. The T1 theorem: This is one of the most fundamental results in the theory of Singular Integrals, due to G. David and J. L. Journe. We are going to give a complete proof which uses some modern ideas of Nazarov, Treil and Volberg.
2. Removability for bounded analytic functions and the Cauchy transform: We are going to see how real-variable methods from harmonic analysis can be used in order to answer a problem from complex analysis dating back to 1890.
3. Kakeya and Besicovitch sets: or how small can a set be containing a needle in every direction? We are going to explore techniques from harmonic analysis which are currently employed in the study of Kakeya and Besicovitch sets.

If time permits we are also going to cover the following topics as well:

1. Bochner's theorem and Fourier analysis on locally compact abelian groups.
2. Application of Fourier analysis to dispersive PDE, especially to Schrodinger and KdV equations.

Although the course can be considered as a sequel to 5140, and definitely some basic knowledge of Fourier analysis will be helpful, 5140 will not be a prerequisite. There will be no textbook, I will type and distribute my notes.

Instructor: Vasilis Chousionis

Prerequisites: MATH 5111 or solid understanding of measure theory is required

Offered: Spring

Credits: 3

Sections: Spring 2017 on Storrs Campus

PSCourseID Course Sec Comp Time Room Instructor
24846 5141 001 Lecture MWF 10:10:00 AM-11:00:00 AM MONT 314 Chousionis, Vasilis