Also try to partially answer your second question here, I would say:
1: Differential Equations including Ordinary Differential Equations, Difference Equations, Delay Differential Equations, Integral-Differential or Integral-Difference Equations, 2: Dynamical Systems; 3: Game Theory; 4: Numerical Analysis; 5: Programming techniques: MatLab, Maple, Python, etc.
In the workshop I recently attended, I heard talks about using Algebraic Geometry and Lie Theory in this field as well.
