Quantifying evolutionary uncertainty: This research introduces the wald space, a length space for phylogenetic trees, necessary for quantifying uncertainty in the statistical analysis of evolutionary relationships. The study formally introduces the wald space, examining its topology and structure in detail. It is shown that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterisation of cube boundaries, it is demonstrated that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, it is proven that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fréchet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.
Published in the Journal of the London Mathematical Society, this paper contributes to the journal's focus on advanced mathematical research. By introducing the wald space and analyzing its properties, the study provides a mathematical foundation for understanding phylogenetic trees, aligning with the journal's scope.