The Lumenary
Un mapa completo de las matemáticas desde preescolar hasta el primer año de posgrado, con caminos opcionales hacia la investigación. Las lecciones se escriben paso a paso, y cada paso se comprueba con cuidado.
Temas: 200
Rutas sugeridas a través del grafo. Los programas de posgrado varían, así que son puntos de partida y no requisitos.
School learners from kindergarten to the end of secondary school, and the teachers and families supporting them.
A steady route through school mathematics: counting and collections, arithmetic and fractions, sets and integers, functions and algebra, geometry and trigonometry, probability, and introductory calculus. Optional Wonder Trails branch off at every stage but are never required.
Secondary-school learners who are ready to move faster and further, including those preparing for olympiads or early university study.
Builds proof and structure early: logic and proof, countability and Cantor's diagonal argument, composition and structure-preserving maps, a first universal property, then calculus, linear algebra and algebraic structures. Acceleration should deepen understanding, not merely cover topics sooner.
Adults returning to mathematics, or meeting it properly for the first time, at any age and with any prior experience.
Starts from whole-number arithmetic without assuming school knowledge is fresh, and moves at an adult pace through fractions, integers, sets and logic, functions and equations, everyday statistics and probability, and a first experience of proof and composition. No timed work and no prerequisite shame: revisit any topic as often as needed.
Undergraduates studying mathematics as a main subject, and independent learners following an equivalent programme.
A typical core for a mathematics degree: proof, calculus, linear algebra, real and complex analysis, abstract algebra with quotients and Galois theory, topology, number theory, graph theory, a first course in categories and functors, and mathematical writing. Degree programmes differ in order and emphasis; this is one coherent ordering, not the only one.
Learners heading towards theoretical computer science, programming-language theory, verification or proof engineering.
Logic and proof alongside programming, then algorithms, discrete mathematics, Lean, computability and complexity, the λ-calculus and type theory, a categorical view of structure, and formal methods. Formal proofs are introduced gradually, and a machine-checked proof is only claimed when it has actually been checked.
Learners aiming at science, engineering, data, economics, biology or computation, who want mathematics they can use and trust.
Functions and calculus, linear algebra and differential equations, probability and statistics, numerical methods and analysis, partial differential equations, optimisation, mechanics, information theory and mathematical biology. Emphasises modelling assumptions and the difference between exact results, numerical approximations and statistical evidence.
Advanced undergraduates and others preparing for a first year of graduate study in mathematics.
Consolidates the foundations most first-year graduate programmes assume (analysis, algebra, topology and category theory) and then samples the standard first-year courses: measure theory, functional analysis, modules and commutative algebra, representation theory, algebraic topology and smooth manifolds. Graduate programmes differ considerably, so learners should adapt the selection to the programmes and specialisations they are considering.
Graduate students and advanced independent learners preparing to read, write, check and contribute to research mathematics.
Research skills (writing, reading the literature, reproducible computation, experimental mathematics, formalisation and research ethics) together with one illustrative specialisation: from the Yoneda lemma and homological algebra through quivers and path algebras to derived categories and quiver representations. Research directions vary widely, so the specialisation is an example to adapt, and established results are always distinguished from open questions.
Learners drawn to shape, space and continuity: future geometers and topologists, and anyone who wants to see mathematics through the geometer's and topologist's eyes.
Follows the geometric thread from Euclid to current research: vectors and non-Euclidean geometry, surfaces and the Euler characteristic, point-set topology, curves and surfaces, the fundamental group, differential forms and knots, then smooth manifolds, algebraic and differential topology, Riemannian geometry, homotopy theory and characteristic classes, ending at low-dimensional topology. Set theory and category theory appear where geometry needs them — as languages for spaces, maps and invariants — not as replacements for geometric intuition. Programmes differ, so adapt the order and the research end-point (for example symplectic, Kähler or geometric group theory instead).
Learners who want the analyst's view of mathematics — limits, estimates, approximation, differential equations and long-term behaviour — whether heading to pure analysis, applied mathematics or physics.
From functions and calculus through real and complex analysis to the core graduate subjects of measure theory and functional analysis, then onward to harmonic analysis, the modern theory of PDE, dynamical systems and ergodic theory, ending with geometric analysis, where nonlinear PDE answer questions about shape. Each stage pairs rigorous estimates with numerical experiment, always distinguishing evidence from proof.
Learners fascinated by whole numbers, equations and symmetry, heading towards algebra, number theory or arithmetic geometry.
Begins with divisibility and congruences and builds the algebraist's toolkit — groups, rings, quotients, fields and Galois theory, modules and commutative algebra, representation theory — alongside the number theorist's questions about primes and equations. It continues through algebraic and analytic number theory and algebraic geometry to elliptic curves and an orientation to the Langlands programme, separating proved theorems from open conjectures at every step.
Learners who want to understand the mathematics in which physical theories are written — mechanics, relativity, quantum theory and gauge theory — and the physicist's viewpoint on geometry.
Combines analysis, linear algebra and geometry with physical theory: classical mechanics and its symplectic geometry, continuous symmetry through matrix groups and Lie groups, Fourier analysis and functional analysis for quantum mechanics, Riemannian geometry for relativity, and bundles and connections for gauge theory. Physical intuition and mathematical rigour are kept distinct: heuristic arguments are labelled as such, and open problems (such as the Yang–Mills mass gap) are identified honestly.