Fruit flies and moduli: interactions between biology and mathematics

TL;DR

This paper explores how geometry and topology can be applied to biological data, specifically fruit fly wing veins, revealing new mathematical questions and methods in the analysis of biological structures.

Contribution

It introduces novel applications of algebraic topology and geometric probability to analyze biological features, bridging biology and pure mathematics.

Findings

Topological features in fruit fly wing veins analyzed using multiparameter persistent homology.

Sampling from moduli spaces has significant mathematical implications.

Geometric probability on stratified spaces explains phenomena like the sticky Frechet mean.

Abstract

Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.