The perfect picnic project: Sandwiches, eggs, and trees

Speaker: Andrew Francis

Affiliation: University of Western Sydney

Time: Wednesday 15/12/2021 from 16:00 to 17:00

Venue: Zoom Only

Zoom ID: 897 6927 3460 Password: CDMS

Abstract: Persi Diaconis and Susan Holmes showed in 1998 that binary phylogenetic trees on $n$ leaves can be encoded by matchings on a set of $2n-2$ elements, that is, a partition of the set into pairs. In this talk we show how this matching can be extended to a correspondence with unbalanced Brauer diagrams (and, more generally, partition diagrams) that preserve elements of the structure of the tree. We even get what appears to be a new correspondence for the entire set of forests. Diagrams such as the Brauer and partition diagrams have an associated natural action of the symmetric group, and through Green's relations of semigroup theory, we can construct “eggbox diagrams” displaying equivalence class structures. Another semigroup construction, the sandwich product, allows us to define a product on the set of trees, which is given relative to a chosen tree. With this, we can consider various related semigroup substructures, such as the regular subsemigroup, or idempotents, in the context of phylogenetic trees. Joint work with Peter Jarvis (U Tasmania).

Biography: Andrew Francis is a Professor and ARC Future Fellow in the Centre for Research in Mathematics in the School of Computing and Mathematics at UWS. His research interests lie in algebra (finite reflection groups and Hecke algebras), in mathematical biology (evolutionary questions related to bacterial genetics), and in applications of algebra to biology.