The paper establishes non-trivial dimensional thresholds for volume vectors determined by hypergraphs of simplices via a Jacobian method leveraging distance results and a refinement for planar triangles, improving prior bounds.
New bounds on the dimensions of planar distance sets
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The survey organizes topological methods for counting solutions to Allen-Cahn equations and systems, connecting them to minimal hypersurfaces and multi-phase isoperimetric problems via Gamma-convergence.
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On volume vectors determined by hypergraphs in thin subsets of Euclidean space
The paper establishes non-trivial dimensional thresholds for volume vectors determined by hypergraphs of simplices via a Jacobian method leveraging distance results and a refinement for planar triangles, improving prior bounds.
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Survey on topological methods for Allen--Cahn equations and systems
The survey organizes topological methods for counting solutions to Allen-Cahn equations and systems, connecting them to minimal hypersurfaces and multi-phase isoperimetric problems via Gamma-convergence.