Tropical Geometry of Phylogenetic Tree Spaces

Explore the tropical geometry of phylogenetic tree spaces in this 59-minute lecture by Anthea Monod. Delve into the BHV (Billera Holmes Vogtmann) space, a moduli space of phylogenetic trees with applications in computational biology, computer vision, combinatorics, and category theory. Discover the homeomorphism between BHV space and the Grassmannian using tropical geometry, and learn how this tropical structure enhances computational studies. Examine the coincidence between BHV space and the tropical Grassmannian, and gain insights into recent research on the tropical Grassmannian as a metric space. Understand the practical implications of these findings on probabilistic and statistical studies of phylogenetic tree sets. Cover topics such as notation, terminology, phylogenetic tree origins, tree comparison methods, BHV space construction, the BHV metric, tropical geometry examples, statistical challenges, and the tropical metric's stability.

Introduction
Notation and Terminology
Where do phylogenetic trees come from?
Motivation and Overview
Comparing Phylogenetic Trees
BHV Space: A Geometric Construction
The BHV Metric
Defining Phylogenetic Trees as Metrics
Tropical Geometry
An Example: 4 Leaves
Statistical Issues: Stickiness & High Dimensionality
The Tropical Metric
Stability
What else do we know?