Approaches to Bounding the Exponent of Matrix Multiplication

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Survey describes embedding n×n multiplication into group algebras and into commutative association schemes, using the triple product property to reduce to block-diagonal multiplications; constructs (wreath-product) examples that produce exponents 2.48 and 2.41 but fail to reach 2 because large irreducible dimensions persist. Defines S-rank and proves Omega ≤ (3 Omega_s - 2)/2, so proving Omega_s → 2 would imply Omega → 2. Work produces concrete constructions and numeric bounds but the main conjecture remains open.

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" chapters (22)

1. 0:00Introduction
2. 1:11History
3. 2:02Outline
4. 2:33The matrix multiplication tensor
5. 9:08Upper bounds on rank
6. 14:55A different approach
7. 16:44The general approach
8. 21:40The Group Algebra
9. 22:25"Nice basis" embedding
10. 23:18The embedding
11. 25:28How many multiplications?
12. 30:15The triangle construction
13. 37:25Constructions in linear groups
14. 42:06Group algebra approach
15. 46:58General semi-simple algebras
16. 48:00Weighted vs. unweighted MM
17. 52:21Coherent configurations group theory without groups
18. 53:33Coherent configs: examples
19. 55:18Nice basis conditions
20. 55:41Coherent configs vs. groups
21. 57:21Proof idea
22. 57:47Open problems