Zeros of Random Sections on Line Bundles

TL;DR

This paper investigates the expected distribution of zeros in smoothed random sections of line bundles on surfaces, providing a probabilistic understanding of their structure after smoothing in a discrete setting.

Contribution

It introduces a method to predict the distribution of zeros of smoothed random sections of hermitian line bundles on discrete complexes, extending previous smoothing techniques.

Findings

Predicted the expected sum of indices of zeros on faces.

Analyzed the distribution of zeros after smoothing random sections.

Extended smoothing operator analysis to discrete line bundle sections.

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.