Hello from the other side (of Re(s) = 1/2): Arithmetic harmonic analysis
The Fourier transform is a fundamental symmetry of functions on the real line,
intertwining additive and multiplicative structures. It turns out that this symmetry
is not at all unique to R, and can be defined in the exact same way for functions
on the p-adic numbers. We'll describe how this works and see some examples:
in particular, the p-adic "Gaussian" is the indicator function of the unit "interval"!
In John Tate's incredible Ph.D. thesis, he discovered that you can take the Fourier
transform "over all p at once", and used this to prove the functional equation for the
Riemann zeta function and its variants. To do this, we must consider functions over
the ring of adeles: this is a huge topological ring built out of all of the Q_p's and R.
We'll introduce this beast and see how its Fourier transform encodes fundamental
facts of number theory.
We'll construct everything "from scratch", so analysts and algebraists alike should
find something enjoyable here.