Solving $1$-Laplacians of Convex Simplicial Complexes

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Proposal: fast solver for 1‑Laplacians on certain 3D simplicial complexes that reduce to zero-Laplacian problems via dualization. Method: use Helmholtz decomposition to split demands into potential + circulation, find an initial face flux by exploiting collapsibility of convex triangulations (Whitehead), then optimize circulations with a KOSZ‑style iterative energy minimization and call existing nearly‑linear graph Laplacian solvers. Works under assumptions: convex triangulated 3‑ball, β2=0 and collapsible subcomplexes; practical extension beyond those restrictions remains open.

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" chapters (15)

1. 0:00Introduction to laplacians
2. 1:33Graph laplacian solvers
3. 3:24Higher dimensional topology
4. 5:11Regression and simplices
5. 7:43Helmholtz-Hodge motivation
6. 10:23The zero laplacian
7. 13:52Terminology and chains
8. 17:07Flow decomposition
9. 22:22The Kosc-style algorithm
10. 25:09Decomposition and projection
11. 28:10Solving via minimization
12. 34:13Dualizing equations
13. 38:07Collapsibility and complexes
14. 44:57Spheres and topologies
15. 47:20Social networks and closure