Non-vanishing of twists of GL(4) L-functions

I will discuss recent work with Maksym Radziwill in which we show that for any fixed tempered cuspidal representation π of GL(4) over the rationals, there exist infinitely many primitive characters χ such that the twisted L-function L(s,π×χ) is non-vanishing at the central point s=1/2. I will focus on the proof, which involves a mix of ideas.