### MATH 5141: Abstract Harmonic Analysis

**Description:** Harmonic analysis on various spaces such as Euclidean spaces, and abelian and non-abelian locally compact groups. Pontryagin duality, the Peter-Weyl theorem, various Fourier transforms and connections to unitary representation theory. With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5111.

**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:

- 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.
- 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.
- 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:

- Bochner's theorem and Fourier analysis on locally compact abelian groups.
- 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 |