Hendy, Penny and Steel made a startling discovery in the early nineties, when considering the relationship between branch lengths of a phylogenetic tree and the site patterns that could arise from sequences evolving on that tree.

Each site in an alignment can suggest a split or "bipartition" of the extant taxa of a phylogenetic tree into two sets, and the complete alignment constitutes a spectrum of such splits.

Correspondingly, each phylogenetic tree can be thought of as a collection of compatible splits.

So what is the relationship between the two?

Hendy et al. found an elegant way of converting between the relative frequencies of splits supported by an alignment, and branch lengths of the underlying tree.

This conversion uses a specialised family of Hadamard matrices (such matrices are all square and with entries that are all ±1).

Moreover, this conversion, known as the Hadamard Conjugation, was possible even without choosing a particular tree – rather a remarkable feat – and so enabled researchers to get estimates of branch lengths without conducting a tree search.

Although the computational complexity of the conjugation is exponential in the number of taxa in the tree, this conjugation has led to several significant advances in phylogenetic inference.

This talk will sketch the Hadamard conjugation, and describe how it has influenced phylogenetic estimation over the last two decades.