Some Geometric Perspectives on Combinatorics: High-Dimensional, Local and Local-to-Global II

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Random d‑dimensional simplicial complexes are sampled by keeping the full (d−1)-skeleton and choosing each d‑face independently with probability p; studying the inclusion matrix over GF(2) gives concrete thresholds for homology and analogues of connectivity. For d=2 the vanishing of nontrivial left kernel (no odd intersections) appears at p ≈ 2 ln n / n, proved by choosing a canonical coset representative that is a graph minority in every cut. Collapsibility has a different threshold ~αd/n, and experiments reveal a striking sharp rank‑drop where new faces suddenly create dependencies.

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" chapters (24)

1. 0:00High-dimensional combinatorics intro
2. 1:00Simplicial complexes as objects
3. 6:25Studying simplicial complexes
4. 8:12Combinatorics impact on fields
5. 10:46Random complexes and thresholds
6. 13:15Erdős–Rényi model recap
7. 19:00Connectivity in complexes
8. 22:54Linear algebra for connectivity
9. 30:10Homology trivial solutions
10. 35:26Vanishing first homology threshold
11. 40:30Homology solutions via graphs
12. 46:46Key lemma in proof
13. 50:24Phase transition at p=1/n
14. 55:13Cycles and higher analogs
15. 1:01:11Collapsibility in higher dims
16. 1:03:05Collapsibility thresholds
17. 1:07:53Rank jumps observations
18. 1:13:311D vs 2D cycle differences
19. 1:14:41Hypertrees properties
20. 1:17:09Almost trees in higher dims
21. 1:19:05Algebraic structures higher dims
22. 1:21:35Expander graphs higher dims
23. 1:23:02Algorithmic implications
24. 1:24:26Asymptotic phenomena