Abstract: I will tell three unrelated stories describing new mysteries occurring somewhere inbetween Analysis and Number Theory. (1) The Poincare inequality is a cornerstone of mathematical physics (and related to the behavior of vibrating membranes/plates). I will present a curious improvement on the Torus that has a strong number theoretical flavor - even Fibonacci numbers appear. (2) If the Hardy-Littlewood maximal function of f(x) is “easy" to compute, then f(x)=sin(x). This weird definition of sin(x) has applications in delay-differential equations. One would think that any characterization of sin(x) would be straightforward to prove, and yet the only proof I could find of this one 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, giving rise to surprising pictures.