I will tell three unrelated stories describing new mysteries
occurring somewhere in between Analysis and Number Theory. (1) The Poincare
inequality is a cornerstone of mathematical physics (and related to the
behavior of vibrating plates). I will present a curious improvement on the
Torus that has a strong number theoretical flavor (even Fibonacci numbers
appear). (2) If a function f(x) has the property that its Hardy-Littlewood
maximal function is pretty easy to compute, then the function is sin(x).
This gives a pretty weird definition of sin(x) and has applications in
delay-differential equations. One would think that any characterization of
sin(x) is easy to prove - the only proof I could find relies on a fantastic
miracle to occur (and transcendental number theory). (3) An old integer
sequence (1,2,3,4,6,8...) defined by Stanislaw Ulam in the 1960s turns out
to have very strange properties that may hint towards a new mechanism in
additive combinatorics (and creates very surprising pictures).