Exploring the Poincaré group, its Casimir operators, and Wigner's classification of elementary particles by mass and spin/helicity.

Decomposing the Lorentz algebra into su(2) ⊕ su(2), classifying representations by (j+, j-), and understanding Weyl, Dirac, and Majorana spinors in relativistic field theory.

Exploring the Lorentz group O(3,1), its disconnected components, its defining invariant metric, and its fundamental representations in relativistic physics.

Building the language of Lie groups and their representations from the ground up — generators, structure constants, representations, Casimir operators, and the exponential map.

We derive the Faddeev–Popov identity using a discrete warm-up, insert it into the path integral, extract the gauge-fixed photon propagator, and ask: what other gauge-fixing terms are allowed?

Why you can't just write down the path integral for a gauge theory and call it a day — the operator you can't invert, the orbits you can't avoid, and the geometric picture that makes gauge fixing click.

Decomposing the fluctuations of massive vectors and massless 2-forms to determine their physical degrees of freedom and observational signatures.

Applying the cosmological principle to massive vectors and massless 2-forms to see if they can drive the expansion of the Universe.