Phylogenetic Algebraic Geometry

TL;DR

This paper introduces phylogenetic algebraic geometry, exploring algebraic varieties from evolutionary trees, connecting classical geometric objects with new models for larger trees, and highlighting open research problems.

Contribution

It provides a comprehensive introduction to the field and identifies new algebraic structures arising from larger phylogenetic trees.

Findings

Classical geometric objects are recovered for small trees.

Larger trees lead to new, largely unexplored algebraic varieties.

The paper presents numerous open problems for future research.

Abstract

Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers.