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
April 17, 2026
Interstellar: a brane-bulk reading
A physics reading of Nolan's Interstellar (2014) in which Kerr geometry, Randall-Sundrum II brane-bulk coupling, and M-theory singularity resolution collapse the film's apparent plot holes into consequences of one maintained parameter: Gargantua's spin.
April 15, 2026
From RVE to Mesh: A Pipeline for Heterogeneous Continua
A single pipeline from microstructure to discrete solver: mean-field homogenisation on a representative volume element produces an SPD permeability tensor field, which induces a Riemannian metric, whose Hodge star discretises the Laplace-Beltrami operator, and whose scalar curvature drives adaptive remeshing.
April 14, 2026
Analysis on Manifolds V: Stokes’ Theorem
The generalised Stokes theorem proved in full, recovering the fundamental theorem of calculus, Green’s theorem, the divergence theorem, and the classical Stokes theorem as special cases. Hodge decomposition with complete Sobolev proof. Harmonic representatives, Betti numbers, and the de Rham isomorphism. The conclusion of a five-part lecture series.
April 13, 2026
Analysis on Manifolds IV: Integration
Forms as antiderivatives, oriented manifolds, manifolds with boundary, partitions of unity, integration of k-forms over k-submanifolds, the change-of-variables theorem via pullback, the Riemannian volume form, period integrals, the Mayer-Vietoris sequence, and the full de Rham cohomology of spheres, tori, and surfaces. Part IV of a five-part lecture series on differential forms and the generalised Stokes theorem.
April 13, 2026
Analysis on Manifolds III: Differential Forms
Smooth manifolds, tangent and cotangent spaces, differential forms as smooth sections of Λᵏ(T*M), the exterior derivative, pullback along smooth maps, and the recovery of grad, curl, and div as the exterior derivative in ℝ³. Part III of a five-part lecture series on differential forms and the generalised Stokes theorem.
April 12, 2026
Analysis on Manifolds II: Exterior Algebra
The algebraic machinery behind differential forms: dual spaces, multilinear alternating maps, the wedge product, bases and dimension of Λᵏ(V*), determinants as top forms, the interior product, and the Hodge star. Part II of a five-part lecture series on differential forms and the generalised Stokes theorem.
April 4, 2026
Numerical Analysis via Discrete Exterior Calculus
A self-contained reconstruction of numerical analysis through discrete exterior calculus: simplicial complexes, cochains, the discrete Hodge star, and the Hodge Laplacian, applied to quantum mechanics, computational electromagnetics, and fluid dynamics.
March 28, 2026
Analysis on Manifolds I: Analysis on ℝⁿ
A self-contained, proof-based reconstruction of single and multivariable analysis from first principles: topology of ℝⁿ, the derivative as a linear map, the chain rule, and the inverse and implicit function theorems. Part I of a five-part lecture series on differential forms and the generalised Stokes theorem.
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.