Incidence Geometry and Applications to Theoretical Computer Science

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Shows quantitative Sylvester–Galai bounds: δ-configurations have dimension O(1/δ) (tight), derived from rank bounds for design matrices over characteristic zero and large characteristic. Gives a stable version under bounded pairwise distances that forces approximate low-dimensionality. Proves a new lower bound for three-query locally correctable codes over the reals (n ≥ k^{2+ε}, ε ≈ 1/1000, equivalently k ≤ n^{0.499}) by proving a clustered triple structure—roughly √n orthogonal clusters—and using Barak's isotropic transform plus an improved random-restriction argument.

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" chapters (16)

1. 0:00Intro
2. 1:11Points in Complex space
3. 2:38Quantitative SG
4. 4:55Incidence Theorems from Rank Bounds
5. 7:31Design Matrices
6. 8:16Main Theorem: Rank Bound
7. 10:12Stable Sylvester-Gallai Theorem
8. 16:42Dirac-Motzkin conjecture
9. 18:12(Linear) Locally Correctable Code
10. 20:56Equivalent Geometric Definition
11. 23:04Locally correctable codes
12. 27:113 query LCCs over R: Geometric question
13. 36:58Structure theorem: Main idea
14. 39:35Improved Random Restriction
15. 41:47Summary
16. 42:45Future Directions