High-Dimensional Expanders 4

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Claim: global hypercontractivity and small-set expansion for epsilon-global functions on the p-biased cube extend to certain high-dimensional expanders by treating partite complexes as approximate products. Tools: Efron–Stein decomposition, localization to marginal bipartite graphs, and the trickling down theorem to lift codimension-2 expansion; key notion is a gamma product (pairwise and conditioned marginals are gamma-expanders). Main quantitative step bounds marginal operator eigenvalues (lambda ≤ gamma/(1−gamma)), but error terms multiply with dimension so the method needs very small gamma and currently works best for strong algebraic HDX constructions.

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" chapters (10)

1. 0:00Introduction and background
2. 2:06Hypercontractivity overview
3. 5:54Product spaces challenges
4. 8:33Failure of naive extensions
5. 14:26Global functions definition
6. 21:19HDX as product structures
7. 30:17Partite complex definitions
8. 34:36Gamma product theory
9. 41:45Proof of spectral bounds
10. 1:01:16Conclusion and discussion