Purpose: This is a graduate topics course in Fourier Analysis. The purpose of this course is to discuss some recent results related to Erdos-Falconer type problems in discrete, continuous and arithmetic settings, look for commonalities and differences among those results and then try to understand to what extent they can be extended to the setting of locally compact abelian rings.

Structure: We shall meet twice a week for an hour and a half, though on some occasions I expect the discussion to last much longer. During the first meeting of each week we shall discuss the known results in finite field, p-adic, discrete and continuous settings. The second meeting of each week will be dedicated to the study of the background material pertaining to locally compact abelian groups. We shall mainly use "Fourier Analysis on Groups" by Walter Rudin.

Prerequisites: The main prerequisite is willing to work hard and stay constantly engaged in the course. This is not a kind of a course where you can just sit back and see what the proverbial cat drags in. Everybody will be expected to participate in class discussion and everybody will be asked to give a presentation from time to time.

As far as technical prerequisites go, everybody is expected to have a solid grasp of Tom Wolff'a Harmonic Analysis, though we shall definite cover some of the topics in the beginning of the course. The participants are also expected to have basic grasp of undergraduate algebra and number theory.

Class notes: I will continually post updated notes for the class on this paper and I encourage you to start reading those long before the course begins in January. As you will see, the notes have two parts, consistent with the description of the structure of the course above. As the notes evolve, feel free to ask questions, point out corrections and send in suggestions.

Background materials: I will post various notes on the material we should know in principle, but may not remember in practice.

Notes on the partition of unity

Notes on finite field additive characters

Research papers we are likely to cover in the course:

On the size of Kakeya sets in finite fields, by Zeev Dvir

A sum product estimate in finite fields, and applications, by J. Bourgain, N. Katz and T. Tao

Sums and products in finite fields: an integral geometric viewpoint, by D. Hart and A. Iosevich

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture, by D. Hart, A. Iosevich, D. Koh and M. Rudnev

Erdos distance problem in vector spaces over finite fields, by A. Iosevich and M. Rudnev