What mathematical skills are needed to learn Lagrangian Field Theory (building up to QFT)?

I want to start teaching myself Lagrangian Field Theory. I can do multivariable calc, tensor calc, Lagrangian mechanics, and some calculus of variations. Are there other math fields I should study before diving in to field theories? I’m hoping to work my way up to QFT.

I would recommend also being able to understand:

• partial differential equations. Some particular PDEs of interest are the wave equation, the Klein–Gordon equation, the Dirac equation, the Broglie wave equation, Maxwell's equations, and the Weyl equation.
• special relativity. Classical and quantum field theories are done in the space-time of special relativity, and fields are required to be invariant under Lorentz transformations.
• differential geometry. If you're going to get serious, you should learn differential geometry up to at least principal bundles, connections and curvature, as well as Lie groups and algebras, since modern field theories are best expressed in the language of differential geometry.
• representation theory. Learn at lease some very basics of it, since this area is heavily drawn upon in both classical and quantum field theories, and you'll see what I mean when you dip your toes into field theories.
• basic physics. Finally, it goes without saying that if you want to learn classical and quantum field theories, you need to start from first principles with those big thick 1500 page physics books, and learn physics in parallel with the mathematics.