High-Dimensional Expanders 2

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" summary

Proves a concrete local‑to‑global claim: link expander ⇒ random‑walk expander by rewriting the upper walk as an expectation over links, bounding Mi+⟨f,f⟩ by UD⟨f,f⟩ plus λ||f||^2 via each link’s adjacency Ar and its spectral gap. Applies to the matroid/basis‑exchange walk on spanning trees, showing the top links are multipartite with second eigenvalue ≈0 so the tree walk mixes in poly(n) steps. Develops F2 coboundary maps, shows coboundary = cuts (i=0) and gives a nontrivial cocycle not a coboundary example; discusses connections and counterexamples between spectral and coboundary notions.

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" chapters (6)

1. 0:00Link expansion introduction
2. 2:55Proof of link expansion
3. 12:06Basis exchange random walks
4. 20:11Posets and expansion
5. 23:42Co-boundary expansion basics
6. 47:28Property testing and conclusion