A rigorous treatment of renormalization group flow: Wilsonian RG, beta functions, fixed points, scaling dimensions, the Callan-Symanzik equation, asymptotic freedom, and the meaning of scale in quantum field theory.
A rigorous treatment of gauge symmetry: local invariance, covariant derivatives, non-Abelian field strength, Yang-Mills action, BRST quantization, instantons, anomalies, and spontaneous symmetry breaking.
A rigorous geometric treatment of general relativity: metric connections, curvature tensors, geodesics, Einstein field equations from variational principles, the initial value problem, and the limits of classical gravity.
A rigorous treatment of quantum field theory: canonical and path integral quantization, correlation functions, perturbative and non-perturbative methods, renormalization, and the limits of the framework.
A rigorous treatment of the action principle: variational calculus, boundary terms, Noether's theorems, constrained systems, and the route to quantization.
A rigorous analysis of the conceptual and mathematical obstructions to unifying quantum field theory, general relativity, and cosmology.