Lesson 252
Topology & Topological Data Analysis
topology · homology · persistence · TDA
1:00How topology counts holes in shapes — and how persistent homology turns that into a data-analysis tool that finds real structure in noisy point clouds.
By the end, you can
- Explain what a homeomorphism is and why a mug and a donut are topologically identical.
- Compute the Euler characteristic for simple polyhedra and state the expected value for a sphere and a torus.
- State the Betti numbers (b₀, b₁, b₂) for a circle, sphere, and torus.
- Describe the Vietoris–Rips filtration rule and trace how b₀ and b₁ change as ε grows on a noisy ring.
- Interpret a persistence barcode and diagram: identify which features are signal and which are noise.
- Explain why the stability theorem makes TDA robust to noise.
- Distinguish persistent homology from the Mapper algorithm and name a real-world application of each.
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