LAUSR

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral relationships between species. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition -- techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, in relation to the graphical structures (trees and networks) which form the search space for inferring evolutionary trees. Aspects presented, include relating model classes to relevant matrix Lie algebras, as well as manipulations with group characters to enumerate various natural polynomial invariants, for identifying robust, low-parameter quantities for use in inference. Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the `tangle') can be directly imported into the formalism -- for models with binary character traits, and triplets of taxa. In other cases new objects appear, such as the remarkable quintic `squangle' invariants for quartet tree discrimination and DNA data, with their own unique interpretation in the phylogenetic modeling context.