Smooth Selection Principles, Simplicial Complexes, and Everything in Between

Abstract

This thesis is divided in three chapters, each of them completely self-contained, detailing the progress done in three projects.

In the first chapter, we develop efficient algorithms for the approximate smooth selection principle, extending work previously done by Fefferman, Israel and Luli [\emph{Geometric and Functional Analysis} 26.2 (2016): 422-477]. We present complete proof of correctness and efficiency. This is the longest (by far) chapter of the thesis and includes full technical details of the algorithms. We encourage the reader to read the pedagogic explanation of the algorithm in the introduction of the chapter.

In the second chapter, we explore contagion and opinion dynamics on simplicial complexes, extending thus the theory of complex contagion. We present some analytical methods, computational algorithms and inference tools.

The third chapter is fully experimental, and we study the effect of the shape of a domain on the solutions to competition-diffusion equations. It includes some work in progress on a software package to assist conservation ecologists in the design of conservation areas.