Mathematics

From first steps to doctoral study

Start slowly. Each stage builds on the previous one; later branches have their own prerequisites. The foundation lessons are available now. The remaining curriculum is a plan for future lessons, not a complete course.

1. 1. First steps · FOUNDATION LESSONS AVAILABLE

   Before you start: No previous mathematics study.

   Quantities, arithmetic, fractions, decimals, signed numbers, notation, ratios, and introductory equations.

   Ready to continue when you can: Explain a calculation in words and check its answer using a second method.

   Start the foundation lessons →
2. 2. School foundations · PLANNED

   Before you start: Comfort with first-step arithmetic and symbols.

   Percentages, powers and roots, algebra, inequalities, geometry, coordinate graphs, trigonometry, and introductory statistics.

   Ready to continue when you can: Solve unfamiliar multistep problems and state units, assumptions, and domain restrictions.

3. 3. Undergraduate core · PLANNED

   Before you start: Algebra, functions, geometry, and trigonometry.

   Single- and multivariable calculus, linear algebra, discrete mathematics, proof methods, differential equations, and probability.

   Ready to continue when you can: Write elementary proofs and connect symbolic, geometric, and numerical solutions.

4. 4. Advanced undergraduate study · PLANNED

   Before you start: Proof methods, calculus, and linear algebra.

   Real analysis, abstract algebra, complex analysis, topology, numerical analysis, and mathematical statistics.

   Ready to continue when you can: Explain why a theorem’s hypotheses matter and construct counterexamples when they fail.

5. 5. Graduate foundations · PLANNED

   Before you start: Advanced undergraduate study in the chosen branch.

   Measure theory, functional analysis, graduate algebra and topology, advanced probability, and partial differential equations as branch-specific options.

   Ready to continue when you can: Solve sustained proof problems and present a rigorous reading of an advanced text.

6. 6. Research specialization · PLANNED

   Before you start: Relevant graduate prerequisites; no requirement to study every branch.

   Choose a branch such as analysis, algebra, geometry, topology, probability, or applied mathematics; follow its literature and methods.

   Ready to continue when you can: Reproduce a published argument or computational result, identify assumptions, and compare competing approaches.

7. 7. Doctoral research practice · PLANNED

   Before you start: A focused specialization and substantial proof or modelling experience.

   Research questions, literature synthesis, conjectures, original proofs or models, seminars, reproducible computation, and thesis writing.

   Ready to continue when you can: Formulate a tractable question, defend a method, and communicate an original contribution under expert critique.

Doctoral-level work involves specialization, original research, and expert critique. This pathway supports that preparation; completing site lessons does not confer a degree or establish research competence.