Mathematics

Nov 23, 2021 Differential Geometry 72 min read

Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics

Vector bundles generalize the tangent bundle, connections on bundles generalize Levi-Civita, and characteristic classes are topological invariants — this is the geometry underlying gauge theory and general relativity.

Nov 21, 2021 Differential Geometry 70 min read

Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar

The Riemann curvature tensor captures all intrinsic curvature information — its contractions (Ricci and scalar curvature) control volume growth, geodesic deviation, and Einstein's equations.

Nov 19, 2021 Differential Geometry 72 min read

Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport

A Riemannian metric lets us measure lengths, angles, and volumes on any smooth manifold — the Levi-Civita connection provides the canonical notion of parallel transport and geodesics.

Nov 17, 2021 Differential Geometry 62 min read

Differential Geometry (9): Integration on Manifolds and Stokes' Theorem

Stokes' theorem — the fundamental theorem of calculus on manifolds — unifies Green's, Gauss's, and the classical Stokes' theorems into one elegant statement.

Nov 15, 2021 Differential Geometry 64 min read

Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds

Differential forms unify gradient, curl, and divergence into a single framework — the exterior derivative d and wedge product turn calculus coordinate-free.

Nov 13, 2021 Differential Geometry 64 min read

Differential Geometry (7): Vector Fields, Flows, and the Lie Bracket

Vector fields generate flows — one-parameter families of diffeomorphisms. The Lie bracket measures the failure of flows to commute, leading to Frobenius integrability.

Nov 11, 2021 Differential Geometry 62 min read

Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces

Manifolds free geometry from ambient space — charts, atlases, and smooth structure let us do calculus on spaces that don't live in R^n.

Nov 9, 2021 Differential Geometry 62 min read

Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology

The Gauss-Bonnet theorem connects total Gaussian curvature to the Euler characteristic — a stunning bridge between local differential geometry and global topology.

Nov 7, 2021 Differential Geometry 68 min read

Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics

Gauss's Theorema Egregium reveals that Gaussian curvature depends only on the first form — geodesics are the 'straight lines' of curved surfaces, minimizing arc length locally.

Nov 5, 2021 Differential Geometry 64 min read

Differential Geometry (3): The Shape Operator — Curvature of Surfaces

The Gauss map and shape operator capture how a surface bends in space — principal, Gaussian, and mean curvatures classify every point as elliptic, hyperbolic, or parabolic.

Nov 3, 2021 Differential Geometry 66 min read

Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements

Regular surfaces, coordinate patches, the tangent plane, and the first fundamental form — how to measure lengths, angles, and areas on a surface without leaving it.

Nov 1, 2021 Differential Geometry 64 min read

Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame

Parametrized curves, arc length, curvature, torsion, and the Frenet-Serret apparatus — the complete local theory of space curves.

Oct 23, 2021 Functional Analysis 70 min read

Functional Analysis (12): Functional Analysis in Action — PDE and Quantum Mechanics

Lax-Milgram for elliptic PDE, variational methods, quantum observables as self-adjoint operators, and Stone's theorem — where the abstract theory meets concrete applications.

Oct 21, 2021 Functional Analysis 72 min read

Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions

Distributions extend the notion of function to handle derivatives that don't exist classically — Sobolev spaces provide the right setting for weak solutions to PDE.

Oct 19, 2021 Functional Analysis 76 min read

Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions

C₀-semigroups provide the abstract framework for evolution equations — the Hille-Yosida theorem characterizes which operators generate well-posed dynamics.

Oct 17, 2021 Functional Analysis 68 min read

Functional Analysis (9): Unbounded Operators — When Boundedness Fails

Closed operators, the distinction between symmetric and self-adjoint, deficiency indices, Friedrichs extension, the spectral theorem for unbounded self-adjoint operators, and Stone's theorem.

Oct 15, 2021 Functional Analysis 68 min read

Functional Analysis (8): Spectral Theory — Decomposing Operators

The spectrum generalizes eigenvalues to infinite dimensions — the spectral theorem for bounded self-adjoint operators and continuous functional calculus give us a complete decomposition.

Oct 13, 2021 Functional Analysis 62 min read

Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions

Compact operators are limits of finite-rank operators and inherit much finite-dimensional spectral behavior — the Fredholm alternative and spectral theorem for compact self-adjoint operators.

Oct 11, 2021 Functional Analysis 72 min read

Functional Analysis (6): Bounded Linear Operators and the Big Theorems

The Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem — three consequences of completeness that constrain how operators can behave.

Oct 9, 2021 Functional Analysis 68 min read

Functional Analysis (5): Weak and Weak-* Topologies — When Norm Convergence Is Too Strong

Norm topology is too fine for many purposes — weak and weak-* topologies provide the compactness results that make optimization and PDE theory work.

Oct 7, 2021 Functional Analysis 76 min read

Functional Analysis (4): Dual Spaces and the Hahn-Banach Theorem — Taming Linear Functionals

The Hahn-Banach theorem guarantees enough continuous linear functionals exist to separate points — the foundation for duality theory in functional analysis.

Oct 5, 2021 Functional Analysis 68 min read

Functional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions

Inner products give infinite-dimensional spaces geometric structure — orthogonality, projections, and the Riesz representation theorem make Hilbert spaces the analyst's paradise.

Oct 3, 2021 Functional Analysis 78 min read

Functional Analysis (2): Normed Spaces and Banach Spaces

Norm axioms, classical examples, equivalence of norms in finite dimensions, completeness and why it matters, Schauder bases, quotient spaces, and the role of separability.

Oct 1, 2021 Functional Analysis 70 min read

Functional Analysis (1): Metric Spaces — Distance, Convergence, and Completeness

From the real line to infinite-dimensional function spaces: why completeness is the dividing line.

Sep 23, 2021 Abstract Algebra 68 min read

Abstract Algebra (12): Algebra in the Wild — Cryptography, Coding Theory, and Beyond

From RSA encryption to error-correcting codes to particle physics — abstract algebra's most powerful real-world applications, and where to go next.

Sep 21, 2021 Abstract Algebra 62 min read

Abstract Algebra (11): Category Theory — The Language of Mathematical Structure

Categories, functors, and natural transformations provide a universal language for mathematical structure — and universal properties replace ad hoc constructions with elegant characterizations.

Sep 19, 2021 Abstract Algebra 62 min read

Abstract Algebra (10): Representation Theory — Groups Acting on Vector Spaces

Representing abstract groups as matrices makes them concrete and computable — Maschke's theorem, Schur's lemma, and character theory give us powerful classification tools.

Sep 17, 2021 Abstract Algebra 68 min read

Abstract Algebra (9): Modules — Generalizing Vector Spaces

Modules over rings generalize vector spaces over fields — the structure theorem for finitely generated modules over PIDs unifies the theory of abelian groups and canonical forms.

Sep 15, 2021 Abstract Algebra 70 min read

Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups

The Fundamental Theorem of Galois Theory establishes a perfect correspondence between intermediate fields and subgroups — and settles the ancient question of solvability by radicals.

Sep 13, 2021 Abstract Algebra 66 min read

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems

Algebraic and transcendental extensions, the tower law, minimal polynomials, and splitting fields — the machinery that makes Galois theory possible.

Sep 11, 2021 Abstract Algebra 70 min read

Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition

The division algorithm, irreducibility tests, and the climb from Z to Z[x] to Q[x] — understanding when and why unique factorization holds.

Sep 9, 2021 Abstract Algebra 68 min read

Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture

Adding multiplication to the mix: rings, integral domains, ideals, and quotient rings — the algebraic structures behind number theory and polynomial arithmetic.

Sep 7, 2021 Abstract Algebra 66 min read

Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups

The Sylow theorems give us a systematic way to find and count subgroups of prime-power order — the sharpest tool for classifying finite groups.

Sep 5, 2021 Abstract Algebra 70 min read

Abstract Algebra (3): Quotient Groups and Homomorphisms: The Art of Collapsing Structure

Normal subgroups, quotient constructions, and the isomorphism theorems — how to systematically simplify groups while preserving their essence.

Sep 3, 2021 Abstract Algebra 68 min read

Abstract Algebra (2): Group Actions — How Groups Move Things Around

We formalize how groups act on sets, prove the orbit-stabilizer theorem, derive Burnside's lemma, and count necklaces.

Sep 1, 2021 Abstract Algebra 70 min read

Abstract Algebra (1): Groups — Your First Encounter with Algebraic Structure

From integers to symmetries, we build the formal definition of a group, prove Lagrange's theorem, and compute our first subgroup lattice.