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

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" summary

Counts Latin squares and their higher‑dimensional analogs by reducing to permanents and using Bregman + van der Waerden via permanents to get tight upper bounds; key technical tool is an entropy proof via random row order (Radhakrishnan style) that turns permanents into expected logs of available choices. Shows a 3‑stochastic polytope has many non‑Latin vertices (half‑entry examples), proves an upper bound for d‑dimensional permutations with Tsur‑Lueck, and notes lower bounds and rapid mixing for sampling remain open.

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" chapters (10)

1. 0:00Introduction
2. 2:19High-dimensional combinatorics motivation
3. 7:30Higher-dimensional permutations
4. 12:46Latin squares and permanents
5. 27:15Bounds for Latin squares
6. 33:17Bregman’s theorem proof
7. 47:38Extending to higher dimensions
8. 1:07:35Entropy method
9. 1:11:32Random Latin squares
10. 1:23:41Summary