Zeros of Random Sections on Line Bundles

Sections of line bundles on 2 dimensional surfaces in 3 dimensional space can have many distinct shapes. For practical purposes we prefer smooth sections that are visibly easy to follow. This is why smoothing operators have been developed on discrete surfaces as in the inspirational paper “Globally Optimal Direction Fields” [Knoeppel et al. 2013] that can be applied to any section to return another smoother section. We are interested to make predictions on one aspect of the resulting smoothed section’s structure, namely position of its signed zeros. The zeros are the most noticeable feature of a section where the section values circles around a specific point. The purpose of this thesis is to predict the distribution of the smoothed section’s signed zeros with multiplicity that are given by applying the smoothing operator to randomly generated sections of hermitian line bundles on closed simplicial complexes. This will be done in a discrete setting consequently meaning that we will compute the expected sum of indices on each face. Why and how we do this is this thesis’ purpose to explain.

💡 Research Summary

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The thesis investigates the statistical behavior of zeros (also called “singularities” or “critical points”) of sections of Hermitian line bundles defined on closed, two‑dimensional simplicial complexes embedded in ℝ³, after those sections have been smoothed by a heat‑kernel operator. The author’s motivation stems from computer graphics and geometry processing, where direction fields (or complex‑valued sections) are often smoothed to improve visual readability, yet the location and sign (index) of their zeros remain a crucial feature for both aesthetics and topological guarantees (e.g., the Poincaré‑Hopf theorem).

Geometric and Algebraic Setup
A simplicial complex M = (V, Σ) consists of n vertices and a set of edges and triangular faces. Over each vertex i a one‑dimensional complex Hermitian space L_i is placed, forming a discrete Hermitian line bundle L → M. A discrete connection η assigns a unitary map η_{ij}: L_i → L_j to each oriented edge, while a closed real‑valued 2‑form Ω records the net rotation around each face: η_{ki} ∘ η_{jk} ∘ η_{ij} = e^{iΩ_{ijk}} Id. A section φ ∈ Γ(L) is a choice of a vector φ_i ∈ L_i for each vertex; after fixing a non‑vanishing reference section X, every φ can be written as φ_i = z_i X_i with complex scalars z_i ∈ ℂⁿ. Hence the space of sections is identified with ℂⁿ, equipped with a Hermitian inner product weighted by vertex areas.

Discrete Laplacian and Smoothing
The cotan‑Laplacian Δ is defined in the standard way using the cotangents of the two angles opposite each interior edge. Δ is symmetric, positive semidefinite, and has a real spectrum {λ₁ ≥ … ≥ λ_n ≥ 0}. The smoothing operator is the heat kernel S_t = exp(tΔ), t > 0, which damps high‑frequency components of a section. Because Δ acts linearly on ℂⁿ, S_t is a matrix exponential that can be applied directly to the coefficient vector z.

Zeros, Rotation Form, and Index
A zero of a section is a vertex where φ_i = 0, but in the discrete setting zeros may also lie inside a face. To capture this, the author introduces a rotation 1‑form ω on each oriented edge: ω_{ij} = arg(η_{ij} φ_i / φ_j). The index of a face f is defined as
 Ind(f) = (1/2π) ∮_{∂f} ω,
which measures the total winding of the section around the boundary of f. This integer (or half‑integer) encodes both the multiplicity and the sign of any zero(s) inside f.

Random Sections
The random model assumes the complex coefficients z_i are independent standard complex Gaussian variables (mean zero, unit variance). Consequently the random section is a white‑noise field on the mesh. Because the smoothing operator is linear, the smoothed section φ_t = S_t φ remains Gaussian with covariance matrix Σ_t = S_t S_t^*.

Discrete Expectation of the Index
Using linearity of expectation and the definition of ω, the expected index of a face reduces to a combination of entries of the Green’s function G_t = S_t Δ⁻¹ (the pseudo‑inverse of Δ). The author shows that
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