All Writings

Differential form integration, orientability, partition of unity, the Generalized Stokes’ Theorem uniting calculus, and de Rham cohomology groups.

Smooth manifolds, charts and atlases, derivations and tangent spaces, vector fields, Lie brackets, and the algebra of differential forms.

Singular homology, simplicial complexes, nilpotent boundary operators, exact sequences, Betti numbers, cohomology rings, and Poincaré duality on orientable manifolds.

Homotopy equivalence of continuous paths, loop spaces, the fundamental group π₁(X), universal covering spaces, and the Seifert-van Kampen theorem.

Topological spaces without metrics, neighborhoods, open and closed sets, Hausdorff T2 separation, Heine-Borel compactness, and the master theorem of product topologies.

Infinite-dimensional vector spaces over arbitrary fields, algebraic dual spaces V*, the Spectral Theorem, Jordan canonical form, and the universal property of tensor...

Field extensions, algebraic vs transcendental elements, splitting fields, the Galois correspondence bijection, and the Abel-Ruffini theorem.

Ring axioms, ring homomorphisms, prime vs maximal ideals, integral domains, Principal Ideal Domains (PIDs), and Unique Factorization Domains (UFDs).

Abstract symmetries, group axioms, Lagrange’s theorem, normal subgroups, quotient groups, and the fundamental First Isomorphism Theorem.

Formal foundations of modern mathematics, naive set theory contradictions, the Zermelo-Fraenkel axioms with Choice (ZFC), and the cumulative hierarchy.