High-Dimensional Expanders 3

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Constructs a concrete levelwise decomposition of functions on a k‑dimensional simplicial complex by lifting kernels of the down operator and proves uniqueness when the up operator is injective (holds if the complex is a gamma random‑walk HDX with gamma < 1/(k−1)). Uses the U and D averaging operators and random‑walk spectral bounds to define Vki = Uki ker Di, shows approximate orthogonality and that each level is an approx eigenvalue (k−i)/k of the lower walk up to O(k gamma) error. The basis is explicit, works only for strong expansion.

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" chapters (9)

1. 0:00Fourier analysis on HDXs
2. 5:48High-dimensional expanders background
3. 20:48Definition of high-dimensional expanders
4. 29:05Level functions on complexes
5. 36:02Fourier decomposition theorem
6. 47:08Existence of decomposition proof
7. 51:41Approximate eigenbasis property
8. 58:46Summary of results
9. 59:27Q&A beyond simplicial complexes