The idea to embed a Turing Machine in famous problems seems to have popped up a lot in recent years, some examples I’ve noticed :
- Łukasz Grabowski embeds one into integral group rings to tackle a problem of Atiyah on values of Betti numbers ;
- Terence Tao uses the idea in his attack of the Navier-Stokes problem ;
- and yesterday Toby Cubitt (a.k.a. dr-qubit), David Perez-Garcia and Michael M. Wolf also resort to it in relation with the Spectral Gap problem
That last paper does mention Bjorn Poonen’s very interesting essay on the topic of undecidability.