This isn't my field (a.k.a. I'm talking out of my ass), but I can give it my best shot. Usually with QFT you start with locality and unitarity as a sort of starting point. You might hope that you could come up with something much simpler where locality and unitarity come about naturally from the model itself (the goal of physics being to show that "it had to be this way".) QFT tells us how the world works by calculating scattering amplitudes. You throw n things in, and m things come out with some momentum, some spin, some color, etc. The way we started doing this was with Feynman diagrams... these diagrams tend to look like you have some particles banging off of some other particles, and they behave a little that way, but they're actually a notation for integrals in a series that you have to sum over. Hence the genius of the notation... they describe math while looking like something physically relevant. However, they're a little weird in that if you interpreted the diagram literally, you'd find that some of the diagrams involved in an interaction actually aren't physical! One might say that the degree of nonphysicality suppresses that diagram's contribution. Physicists describe these diagrams as "off-mass-shell." If you wish to describe an interaction, you figure out all the diagrams relevant to the interaction, calculate them and sum them. When the article says "tons of pages of calculations", what they mean is you need to sum over a LOT of Feynman diagrams to get an answer.
Now, there's some newish kinda diagram that's gone into use (I think maybe only for N=4 SYM?), which is also used to calculate interaction cross-sections. However, they're written in such a way that they always describe on-shell processes, hence why they're called on-shell diagrams. I think Zvi Bern had something to do with this. These diagrams have an underlying structure... and mathematicians have started writing similar diagrams recently (as in, they ran into a similar structure recently), but instead of the diagrams being to describe interactions, they're used to describe some structure called a positive grassmanian. The positive grassmanian in low dimensions relates to convex polygons... it's a simple thing. This guy had this interesting insight that an interaction crossection corresponds to some sort of volume... and in this case, it corresponds to the volume of this polytope described by the positive grassmanian... or something like that.
The positive grassmanian says nothing at all about unitarity or positivity, nothing about space or time, but it lets you calculate shit. That's very, very cool.
Now, there's some newish kinda diagram that's gone into use (I think maybe only for N=4 SYM?), which is also used to calculate interaction cross-sections. However, they're written in such a way that they always describe on-shell processes, hence why they're called on-shell diagrams. I think Zvi Bern had something to do with this. These diagrams have an underlying structure... and mathematicians have started writing similar diagrams recently (as in, they ran into a similar structure recently), but instead of the diagrams being to describe interactions, they're used to describe some structure called a positive grassmanian. The positive grassmanian in low dimensions relates to convex polygons... it's a simple thing. This guy had this interesting insight that an interaction crossection corresponds to some sort of volume... and in this case, it corresponds to the volume of this polytope described by the positive grassmanian... or something like that.
The positive grassmanian says nothing at all about unitarity or positivity, nothing about space or time, but it lets you calculate shit. That's very, very cool.