Phylogenetic Forest Spaces

Abstract

Tag der mündlichen Prüfung: 04. Oktober 2022 v vi A promising construction that covers the example from above and behaves as biological intuition suggests is our recently introduced Wald Space, cf. Garba et al. (2021a). It is based on the characterization of phylogenetic trees as covariance matrices, which is a byproduct of a generalization of the popular biological substitution models that are used to calculate likelihoods for trees given genetic sequence data, so those substitution models are the backbone of phylogenetic tree estimation. In other words, the Wald Space is a space that is consistent with the tree estimation methods that are currently used, up to the generalizations that have been made. More details can be found in Garba et al. (2021a).In this work, we concentrate on the Wald Space purely from the perspective that it is a mathematical structure and thus we try to enable for a be er understanding of the Wald Space. To this end, we introduce the mathematical structures required: metric spaces, Riemannian manifolds, Riemann strati ed spaces, as well as, for our construction essential, various geometries on the manifold of strictly positive de nite symmetric real matrices. Furthermore, we introduce various possible ways to represent the phylogenetic trees and forests that we consider. Having nished the introduction of the more general and known concepts, we brie y introduce the BHV Space. en we de ne and describe the Wald Space, which is a topological strati ed space, and we investigate its topological features. is part is the core of the thesis. Finally, we equip the Wald Space with a geometry that can be chosen to some degree and nd that these spaces are then Riemann strati ed spaces of type (A). Last but not least, we propose some numerical algorithms to calculate geodesics and distances in the Wald Space equipped with a geometry.ose are not tested in this work, but to some extent in Lueg et al. (2021).