Matrix Factorizations

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" summary

Defines a matrix factorization concretely as matrix factorization = (A,B) with AB = f·Id, proves nontrivial MFs exist for any power series f in the maximal ideal squared by producing A,B from a module of infinite projective dimension, and shows those MFs encode the minimal free resolution which is eventually periodic of period 2. Knörrer periodicity gives that Knörrer periodicity doubles size when adding XY, ADE singularities are exactly the cases with finitely many indecomposables, and quadrics force sizes divisible by 2^{n-1}.

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" chapters (21)

1. 0:00Historical background
2. 3:38Defining matrix factorization
3. 6:15Existence of factorizations
4. 10:28Constructing factorizations
5. 12:50Using free resolutions
6. 15:38Homological algebra basics
7. 20:03Periodic resolutions
8. 21:56Connecting factorization to maps
9. 29:55Factorizations as resolutions
10. 32:25Proving the theorem
11. 39:14Indecomposable factorizations
12. 44:10Non-finite factorization families
13. 47:34Periodicity and variables
14. 50:11Break and Q&A
15. 51:42Sum of squares certification
16. 52:43Adding variables formula
17. 55:09Generalizing complete intersections
18. 56:45Operators and modules
19. 1:07:11Permanent vs determinant
20. 1:17:33Clifford algebra connection
21. 1:23:50Size of factorizations