Abstract: Between thousand to million times per day, your cellphone calculates the Fourier Transform (FT) of certain complex valued functions defined on 0,1,…,N-1, with N large (order of magnitude of thousands and more).
The calculation is done using the Fast Fourier Transform (FFT) - discovered by Cooley--Tukey in 1965 and by Gauss in 1805.

In the lecture I want to advertise a beautiful way—due to Auslander-Tolimieri—to obtain the FFT as a natural consequence of an answer to the following:

Question: How to think on the space of functions on the set 0,1,…,N-1?

Engineers tell us that there are two answers for this question:

(A) as functions on that set, where 0,1,…,N-1 regarded as times; and,

(B) as functions on that set, where 0,1,…,N-1 regarded frequencies;

and then the FT is an operator translating between the two spaces.

In the lecture, I will explain that there is another answer, i.e., a not so well-known third space (C), of arithmetic nature, that also gives an answer to the above question, and then the FFT appears simply as the composition of two operators:
the one translating between spaces (A) and (C), and the one that translates (C) to (B).

Remark: The lecture is prepared to be understood by undergraduate students (from any department) who took basic course in linear algebra.