Automated proving that a program doesn't halt
In contradiction with Gurkenglas' answer, there actually is a community of scientists who work on proving non-termination of programs in various language and formalisms.

An obvious approach would be to check for looping non-termination: for a given program $w$, pick an input $x$ and check to see if the same state is reached twice with the same data.

Non-looping non-termination is obviously more complex, but has been studied, e.g. in Emmes &amp; al Proving Non-Looping Non-Termination Automatically or Endrullis &amp; al Proving Looping and Non-Looping Termination by Finite Automata.