# Zeros of Random Sections on Line Bundles

Bachelor Thesis at TU Berlin (2016). Supervision: Ulrich Pinkall

## 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 seen from Knoppel et al. that can be applied to a
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 sectionโs 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.

## Bibtex

@article{padilla2016randomZeros,

author = {Marcel Padilla},

title = {Zeros of Random Sections on Line Bundles},

journal = {B.S. Thesis},

year = {2016},

month = {06},

publisher = {TU Berlin},

}