Mathematics

The essentials before algebra: fractions, decimals, integers, exponents, and number sense.

Shapes, angles, and space: Euclidean and analytic geometry, circles, and the trigonometry that measures them.

Numerical methods and topology, where computation meets abstract structure.

Groups, rings, and fields: the abstract structures underlying modern mathematics, up to Galois theory.

From linear equations to functions, logarithms, and inequalities: the algebra that unlocks higher math.

Where algebra meets geometry: varieties, curves, schemes, and cohomology.

Topology through algebraic eyes: fundamental groups, homology, cohomology, and fiber bundles.

Limits, derivatives, and integrals: the mathematics of change and the theorems that hold it together.

The mathematics of mathematics: categories, functors, limits, and adjunctions.

The art of counting: permutations, combinations, generating functions, and combinatorial designs.

Calculus over the complex numbers: analytic functions, Cauchy's theorem, and conformal mapping.

Equations that describe change: ODEs, boundary value problems, Laplace transforms, and numerical solutions.

Geometry with calculus: manifolds, curvature, geodesics, and Lie groups.

Smooth manifolds and their global structure: tangent spaces, vector bundles, and de Rham cohomology.

The math of computer science: logic, graphs, combinatorics, number theory, and recurrences.

How systems evolve over time: stability, bifurcations, attractors, and chaos.

Analysis in infinite dimensions: Banach and Hilbert spaces, operators, and distributions.

Networks in the abstract: connectivity, coloring, matchings, and flows.

Decomposing functions into waves: Fourier series and transforms, convolution, and Sobolev spaces.

Vectors, matrices, and linear maps: eigenvalues, determinants, and the workhorse of applied math.

The foundations of reasoning: formal systems, model theory, computability, and Godel's theorems.

The mathematics behind physics: classical and quantum mechanics, relativity, and electrodynamics.

Cross-cutting mathematical topics spanning algebra, analysis, probability, and number theory.

The rigorous theory of size and integration: Lebesgue measure, Lp spaces, and Fubini's theorem.

The study of the integers: divisibility, congruences, primes, and the math behind cryptography.

Matrix computation at scale: factorizations, iterative and Krylov methods, and numerical stability.

Chance made rigorous: probability theory, hypothesis testing, Bayesian methods, and statistical inference.

Calculus made rigorous: sequences, limits, continuity, and Lebesgue integration.

Groups as matrices: characters, irreducible representations, and Lie algebras.

The foundation of mathematics: sets, cardinality, ordinals, the axiom of choice, and the continuum hypothesis.

Mathematics at work: modeling, optimization, numerical methods, and financial and cryptographic math.