University of Canterbury

The accurate modelling of geophysical flows often requires information which is difficult to measure and therefore poorly quantified. Such information may relate to the fluid properties or an unknown boundary condition, for example. The premise of this talk is that when the flow is bounded by a free surface, the deformation of this free surface contains useful information which can be used to infer such unknown quantities. The increasing availability of free surface data through remote sensing using drones and satellites provides the impetus to develop new mathematical methods and numerical tools to interpret the signature embedded in the free surface deformation. This talk will explore two recent examples drawn from glaciology and inspired from volcanology for which free surface data was successfully used to reconstruct an unknown field.

The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m, and the resulting permutation character is the Foulkes character. These characters are the subject of the longstanding Foulkes Conjecture. In this talk, we define a deflation map which sends a character of the symmetric group S_{mn} to a character of S_n. The values of the images of the irreducible characters under this map are described combinatorially in a rule which generalises two well-known combinatorial rules in the representation theory of symmetric groups, the Murnaghan-Nakayama formula and Young's rule. We use this in a new algorithm for computing irreducible constituents of Foulkes characters and verify Foulkes’ Conjecture in some new cases. This is joint work with Anton Evseev (Birmingham) and Mark Wildon (Royal Holloway).

Results that say local information is enough to guarantee global information provide the theoretical underpinnings of many reconstruction algorithms in evolutionary biology. Such results include Buneman's Splits-Equivalence Theorem and the Tree-Metric Theorem. The first result says that, for a collection $\mathcal C$ of binary characters, pairwise compatibility is enough to guarantee compatibility for $\mathcal C$, that is, there is a phylogenetic (evolutionary) tree that realises $\mathcal C$. The second result says that, for a distance matrix $D$, if every $4\times 4$ distance submatrix of $D$ is realisable by an edge-weighted phylogenetic tree, then $D$ itself is realisable by such a tree. In this talk, we investigate these and other results of this type. Furthermore, we explore the closely-related task of determining how much information is enough to reconstruct the correct phylogenetic tree.