The undergrad physics curriculum includes the standard college sequence of calculus and differential equations. Many physics students take more math than this, or today, probably some computer science. We took a "physics" course called "theoretical physics" that filled some gaps in the math sequence, most notably multivariate calculus in cylindrical and spherical coordinates, path integrals, and some stuff like that. Specific upper level physics courses also introduced math topics as needed.

Graduate coursework re-iterated the same "theoretical physics" course, but it was called "mathematical methods." We used a textbook by Arfken, that covered the same multivariate calculus plus vector and tensor math. Finding out what the current textbook du jour is for this course, and perusing the table of contents, would give you a pretty good idea of what to expect.

I didn't go into scattering theory as a specialty, but we covered it in some of the other courses, and I remember doing difficult integrals and differential equations in spherical polar coordinates. The coursework quickly runs you out of problems that can be solved in closed form. I'd be shocked if the real work isn't done today with mostly numerical techniques.

You need to work up to (and through) Quantum Field Theory to understand this.

Thankfully, you don't need to be as good or as thorough as Gerard 't Hooft as you learn all those things!

QFT itself has many prerequisites; you need to be well versed in special relativity, electromagnetism, classical field theory and quantum mechanics to get a hold on it. Each of these areas have their own mathematical prerequisites, which include algebra and calculus, multivariable calculus, pdes, complex analysis and fourier analysis.

General relativity and string theory are also comparative to QFT in scope, prerequisite knowledge and utility in other areas of mathematical physics. You would need to learn these if you wanted to start reading a bit more broadly in scattering amplitudes or in another part of math phys, but you can go quite a long way in scattering amplitudes without too much of them.

As someone who has taught myself significantly more than formal education could, I know how valuable it is to have this sort of knowledge trail accessible. Much appreciated.