Laplacian Eigenvalues, Metric Uniformization, and Incidence Geometry
summary
Proves the graph analogue of uniformization statements by constructing a vertex-weight metric (shortest-path lengths equal vertex-weight sums) whose dual is a flow that minimizes L2 congestion; integral flows let each pair send one path and crossings count as congestion, giving a weighted crossing bound with exponent 5/2 and N^4 scaling in the K=2 complete-graph case. The method builds K disjointly supported functions, uses circle packings for K=2, many Cauchy‑Schwarz steps, and yields spectral bounds plus combinatorial corollaries.
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chapters (10)
- 0:00Introduction to uniformization
- 1:54Eigenvalues and geometry
- 5:30Graph and surface bounds
- 8:55Variational characterization
- 13:52Constructing extremal metrics
- 19:02Circle packing and constraints
- 24:01Flows and crossing numbers
- 33:19The technical theorem
- 37:32Generalizations and open questions
- 41:38Discussion and Q&A