Comparison theorems of phylogenetic spaces and algebraic fans

Rapid developments in high-throughput sequencing have accumulated a wealth of cancer genomics data (44, 12), which has led to the use of phylogenetic methods becoming an important direction in cancer research. In-depth understanding of the topological and metric structures of spaces of phylogenetic trees and phylogenetic networks helps to explain the inter- and intra-patient variability of tumor cells. The geometric structures of phylogenetic spaces also allow comparison between subclones of tumor cells to guide therapeutics discovery (11, 38). Motivated by such research goals, my dissertation proves comparison theorems between phylogenetic spaces that represent evolutionary histories and algebraic fans over simplicial complexes which arise in the moduli space of smooth marked del Pezzo surfaces. I will discuss homeomorphisms and isometries between their projectivized spaces and simplicial complexes formed by root subsystems. Furthermore, embeddings between spaces of phylogenetic trees and networks, and those between the projectivized spaces of phylogenetic trees and networks, are introduced. As a result, algebraic fans that correspond to spaces of phylogenetic trees embed into the fans that correspond to spaces of phylogenetic networks. Likewise, simplicial complexes that correspond to the projectivized spaces of phylogenetic trees embed into the simplicial complexes that correspond to the projectivized spaces of phylogenetic networks. Identifying phylogenetic spaces with structures in mathematics expedites the discovery of the missing pieces in evolutionary models whose counterparts are naturally expected in mathematics, and it also equips investigations in phylogeny with more mathematical tools. The connection between genomic spaces and mathematical structures presented has the potential to address core challenges in oncology with recent advances in algebraic geometry (37), tropical geometry (50, 19, 45), and metric methods (7, 57, 5, 3, 53).