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Zeros of Random Sections on Line Bundles

Marcel Padilla

TL;DR

The work addresses how smoothed sections of discrete Hermitian line bundles on closed 2D simplicial surfaces generate zeros with signed multiplicities, and it analyzes their distribution on faces using a discrete smoothing operator $S_t = e^{-t\Delta}$.Two complementary approaches are developed: a discrete index method based on a rotation form and face indices, and an integral-geometry method via a Kodaira embedding into $\mathbb{CP}^{n-1}$ to relate zeros to signed hyperplane intersections.Key contributions include a discrete index framework for zero density, a transformation-formula-based signed integral geometry result linking densities to the Kähler angle, and a unified convergence analysis showing how smoothing concentrates density in the smallest eigenvalue eigenspace, with practical visualization guidance.These results advance the understanding of zero distributions in discretized line bundles and provide computational routes for predicting and visualizing zeros under smoothing, with implications for discrete spin geometry and direction-field design.

Abstract

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.

Zeros of Random Sections on Line Bundles

TL;DR

The work addresses how smoothed sections of discrete Hermitian line bundles on closed 2D simplicial surfaces generate zeros with signed multiplicities, and it analyzes their distribution on faces using a discrete smoothing operator $S_t = e^{-t\Delta}$.Two complementary approaches are developed: a discrete index method based on a rotation form and face indices, and an integral-geometry method via a Kodaira embedding into $\mathbb{CP}^{n-1}$ to relate zeros to signed hyperplane intersections.Key contributions include a discrete index framework for zero density, a transformation-formula-based signed integral geometry result linking densities to the Kähler angle, and a unified convergence analysis showing how smoothing concentrates density in the smallest eigenvalue eigenspace, with practical visualization guidance.These results advance the understanding of zero distributions in discretized line bundles and provide computational routes for predicting and visualizing zeros under smoothing, with implications for discrete spin geometry and direction-field design.

Abstract

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.
Paper Structure (27 sections, 13 theorems, 110 equations, 10 figures)

This paper contains 27 sections, 13 theorems, 110 equations, 10 figures.

Table of Contents

  1. Abstracts
  2. English
  3. Deutsch
  4. Introduction
  5. Acknowledgments
  6. The Aim of this Thesis
  7. Simplicial Complexes and Hermitian Line Bundles
  8. Sections and Projections
  9. Laplace Operator & Smoothing
  10. Laplace Operator
  11. Section Laplace Operator
  12. Smoothing Operator
  13. Smoothing Applications
  14. Smoothing Convergence
  15. Discrete Index Approach
  16. ...and 12 more sections

Key Result

theorem 1

for some $\mu_i \in \mathbb{C}$. Define $i_{\min} := \min\{ i \in \mathbb{N} \ | \ \mu_i \neq 0\}$. then we have where $[ \ ]$ is the projection map [defined in sec. projection map].

Figures (10)

  • Figure 1: Example of simplicial complex $(V,\Sigma)$ with 4 vertices, 7 edges and 2 faces.
  • Figure 2: Real life comparison of a Zebra's stripe pattern and a zero in a directional field Knoppel:2015:SPS. (Photos courtesy Trisha Shears and André Karwath)
  • Figure 3: A part of a simplicial complex with labels needed to calculate the cotan-Laplace operator entry $\Delta_{ij}$.
  • Figure 4: Heat flow example on planar surface. Height represents the heat. Pictures from heatFlowGif:2007
  • Figure 5: globally optimized direction field example on a bunny. Colored dots indicate the index of a zero. Knoppel:2013:GOD
  • ...and 5 more figures

Theorems & Definitions (38)

  • definition 1
  • definition 2: Discrete hermitian line bundle
  • definition 3: Discrete hermitian line bundle with curvature
  • definition 4: Section
  • definition 5
  • definition 6: Projection map
  • definition 7: Smoothing Operator
  • theorem 1: Smoothing convergence
  • proof : proof
  • definition 8: Discrete rotation form $\xi$
  • ...and 28 more