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 fibre bundles, and fluid dynamics on Riemannian three-manifolds. My interest centres on how geometric structure constrains analytic behaviour: when curvature forces singularities, when topology obstructs global constructions, and how coordinate-free formulations reveal invariances that coordinate-based methods obscure.
Blog
August 9, 2026
Semantic Segmentation: Resolving a Domain and Typing its Pieces
Every segmentation method resolves a domain into pieces and then assigns each piece a type. Reading the field through those two operations puts pixels, point clouds, meshes and shape spaces on one axis, recovers the primal-dual mesh pair on which discrete exterior calculus is built, and gives a precise reason why a message-passing network may place its dual vertices where a discrete Hodge star may not.
July 11, 2026
Quaternions Are the Rotors of Space
Hamilton's quaternions reconstructed on their own terms and then recognised as the even subalgebra of the geometric algebra of space: the imaginaries i, j, k are the basis bivectors, a unit quaternion is a rotor, and the half-angle, the two-to-one cover, and the absence of gimbal lock become plain facts about rotors. A companion to the classical-mechanics article, closing with the native rotation expressions this settles in code.
July 3, 2026
Classical Mechanics from Zero, in Two Languages
Classical mechanics constructed from nothing but an inertial frame, in matrix and linear algebra and in geometric algebra side by side: rotations and rigid-body dynamics without Euler angles, one vector derivative replacing grad, div, and curl, and the Lagrangian and Hamiltonian formalisms in both languages.
June 29, 2026
Configuration-Space Curvature and the Navier–Stokes Singular Set
The curvature of the group of volume-preserving diffeomorphisms is the pressure Hessian. Read against the Caffarelli–Kohn–Nirenberg partial-regularity theory, this geometric identity places the Navier–Stokes singular set where the curvature concentrates or the pressure depletes.
May 31, 2026
Active k-atic Fluids on Riemannian 3-Manifolds
A covariant geometric-calculus formulation of active k-atic hydrodynamics on Riemannian 3-manifolds: one SU(2)/H-hat order-parameter family for polar, nematic, biaxial, and cholesteric order, and a non-equilibrium variational principle that selects braided and knotted disclination structures.
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.