MATH-21 · Advanced · ENGLISH · PLANNED
Build the picture, explain the rule, work an example, then justify it.
Study time is an initial activity estimate, excluding extra reading, revision and extended laboratory or project work. Videos and question bodies have not been produced by adding this outline.
Complete these courses or demonstrate equivalent knowledge. These are recommended prerequisites, not enforced enrolment locks.
YouTube playlist: ARK Mathematics | Topology and Differential Geometry
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Sign in to save this learning pathPractical outcome: Compare shapes using invariants rather than pictures alone.
Explain topological spaces using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain bases using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain product topology using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain quotient topology using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Use the first concepts from this section. Compare shapes using invariants rather than pictures alone. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain homeomorphisms using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain compact spaces using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain connected spaces using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Explain separation axioms using a visual and one worked real-world example. Context: Compare shapes using invariants rather than pictures alone. End with a prediction or check-for-understanding.
Compare shapes using invariants rather than pictures alone. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Topological spaces, Bases, Product topology and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Model a curved surface and compare local distances.
Explain charts and atlases using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain smooth manifolds using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain tangent spaces using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain differential forms using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Use the first concepts from this section. Model a curved surface and compare local distances. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain exterior derivative using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain riemannian metrics using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain geodesics using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain gaussian curvature using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Explain gauss-bonnet introduction using a visual and one worked real-world example. Context: Model a curved surface and compare local distances. End with a prediction or check-for-understanding.
Model a curved surface and compare local distances. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Charts and atlases, Smooth manifolds, Tangent spaces and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Compute loop and hole information for simple spaces.
Explain homotopy using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain fundamental group using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain covering spaces using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain van kampen theorem using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Use the first concepts from this section. Compute loop and hole information for simple spaces. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain simplicial complexes using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain boundary operators using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain homology groups using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Explain euler characteristic using a visual and one worked real-world example. Context: Compute loop and hole information for simple spaces. End with a prediction or check-for-understanding.
Compute loop and hole information for simple spaces. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Homotopy, Fundamental group, Covering spaces and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Combine all three sections: Compute loop and hole information for simple spaces. Submit an original worked artifact, evidence or tests, assumptions and a reflection. Rubric: correctness, method, practical usefulness, reproducibility and communication. Self-reviewed, not certified.
6–12 balanced questions sampling every section of Topology and Differential Geometry, including interpretation, calculation or reasoning, a misconception and a new application. Passing is 70%; review errors before retrying.
Visual constructions, worked problems and proofs. Guided work leads to an independent attempt, with a diagram, explanation, trace, calculation or tested artifact. Question bodies are generated when opened in your private learning path. Programming submissions are stored as text; the platform does not execute submitted code.
Use original or appropriately licensed datasets. Practical science activities use models or supplied data unless a qualified supervisor has reviewed a physical experiment.
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