Tag
Differential Geometry
Differential geometry studies smooth manifolds, tensor fields, connections, and curvature using the tools of calculus and linear algebra. It provides the language for modern physics: general relativity lives on pseudo-Riemannian manifolds, gauge theories on principal fiber bundles, and fluid dynamics on Riemannian three-manifolds. My interest centers on how geometric structure constrains analytic behavior — when curvature forces singularities, when topology obstructs global constructions, and how coordinate-free formulations reveal invariances that coordinate-based methods obscure.
Blog
March 22, 2026
Maxwell's Equations and Gauge Theory: Electromagnetism as a Principal Bundle
Four languages for one theory: vector calculus, differential forms, spacetime algebra, and principal fiber bundles. From the classical field equations to gauge invariance, the Aharonov-Bohm effect, Yang-Mills theory, and Dirac monopoles.
March 10, 2026
Manifolds: The Language of Modern Geometry
A rigorous construction of smooth manifolds from first principles: charts, tangent spaces, Riemannian metrics, curvature tensors, and geometric flows.