Limits, function spaces and the partial differential equations that model fluids, fields and diffusion.
The study of decomposing functions into waves (Fourier analysis) and of the operators that act on them. Its modern core is geometric: restriction, decoupling and Kakeya-type incidence problems.
The attempt to define interacting quantum fields rigorously: axioms, constructions in low dimensions, and a growing toolkit of stochastic analysis. Four-dimensional gauge theory remains out of reach.