nucleation

LIT-1659: Toward a spectral theory of cellular sheaves

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Properties6
StatusActive — read in full 2026-10-27 (NOTE-1233), from arXiv v2; worth reading as the standard introduction to sheaf Laplacians: the kernel is the global sections, the heat flow projects onto them, and a set of spectral graph theory's basics lift with short proofs. Exploratory by its own account: no Cheeger inequality, and the applications are sketches.
Version2
Tagsmathematics, network-science
Cited as “grounds” byCLAIM-471 — Physical interconnection can generate effective constraints on joint behaviour; measurement can estimate correspondences among local descriptions; and adaptive apparatuses can revise those correspondences; gluing expresses compatibility under specified relations, but neither the relations' physical generation nor their adequacy follows from gluing alone, CLAIM-481 — Coupling does not by itself separate time scales: two identical undamped oscillators coupled with stiffness k have normal modes at ω₀ and √(ω₀² + 2k), nearly equal when k ≪ ω₀², and neither decays; separation comes where the coupling dissipates relative differences faster than collective ones, as in ẋ = −ax − k(x − y), ẏ = −ay − k(y − x), whose collective and relative modes relax at rates a and a + 2k, and in a Laplacian flow ẋ = −κLx, where the spectrum of L sets the rates, CLAIM-538 — On a finite graph of contexts whose overlaps carry preferred real offsets a and tolerances δ_e ≥ 0, an assignment within every tolerance, |Dx − a| ≤ δ, exists exactly when no cycle's holonomy exceeds the sum of its tolerances; with every δ_e > 0 the least uniform enlargement of the tolerances that admits one is the largest ratio, over simple cycles, of holonomy to summed tolerance, so the obstruction is certified by a cycle; the feasible allocations of tolerance depend on the offsets only through their class in H¹, so local adjustment can move a mismatch among the overlaps of a cycle and never remove it; and an evaluation that omits one overlap of every witnessing cycle reports a compatibility that does not hold
Cited as “paper” byNOTE-1233 — Hansen & Ghrist — Toward a spectral theory of cellular sheaves
Cited as “source” byTHEORY-1070 — For a weighted cellular sheaf of finite-dimensional vector spaces on a finite cell complex, the kernel of the degree-k Hodge Laplacian is the k-th sheaf cohomology, in degree 0 the space of global sections, and the linear flow ẋ = −Δ^k x converges exponentially to the orthogonal projection of its initial state onto that kernel, each eigencomponent decaying at its own eigenvalue; the inner products on the stalks are part of the data, so the same sections can come with different rates, and the Laplacian does not determine the sheaf, THEORY-1090 — Eliminating an interior vertex of a cellular sheaf on a graph can leave its boundary a space of consistent assignments that no sheaf on the boundary vertices has: the interior can impose a constraint among boundary vertices that no set of pairwise compatibility conditions expresses, so Kron reduction, which turns the Laplacian of a connected graph into a graph Laplacian on any set of boundary vertices, fails for sheaves in general and holds when every vertex stalk has dimension at most one

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Hansen, J., & Ghrist, R. (2019). Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3(4), 315–358. doi:10.1007/s41468-019-00038-7. Preprint: arXiv:1808.01513.

Read in full on 2026-10-27 (NOTE-1233).

Why this is here

A21 made it the mathematical core of the owner’s U20: one operator whose kernel is exact compatibility and whose spectrum sets the relaxation rates, so “A mathematically specified system of compatibility constraints can also determine a hierarchy of dynamical relaxation rates, when the physical coupling realizes an appropriate Laplacian flow.” The record held the same authors’ knowledge-sheaf paper (LIT-232; THEORY-299). It bears on CLAIM-471, CLAIM-473 and CLAIM-478.

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