On the structure of wald space for phylogenetic trees

We further explore the wald space for phylogenetic trees introduced in Tom Nye's talk. As a point set, it sits between the BHV space (Billera, Holmes and Vogtmann, 2001) and the edge-product space (Moulton and Steel 2004). It has a natural embedding in the space of positive definite matrices, equipped with the information geometry. Thus, singularities such as overlapping leaves are infinitely far away, proper forests, however, comprising the "BHV-boundary at infinity", are part of the wald space, adding boundary correspondences to groves (corresponding to orthants in the BHV space). In fact the wald space contracts to the complete disconnected forest. Further, it is a geodesic space, exhibiting the structure of a Whitney stratified space of type (A) where strata carry compatible Riemannian metrics. We explore some more geometric properties, but the full picture remains open. We conclude by identifying interesting and open problems, we deem interesting.

Co-authors Tom Nye, Jonas Lueg, Maryam Garba