Durham University

I will talk about the diffeomorphism classification of 4-manifolds up to 
connected sums with the complex projective plane, and how the resulting 
equivalence class of a manifold can be detected by algebraic topological 
invariants of the manifold.  I may also discuss related results when one 
takes connected sums with another favourite 4-manifold, S^2 x S^2, instead.

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.
 

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.

In this talk we will start by introducing the notion of Siegel-Jacobi modular form and explain its close relation to Siegel modular forms through the Fourier-Jacobi expansion. Then we will discuss how one can attach an L-function to an appropriate (i.e. eigenform) Siegel-Jacobi modular form due to Shintani, and report on joint work with Jolanta Marzec on analytic properties of this L-function, extending results of Arakawa and Murase. 

While it is believed that many particle systems have periodic ground states, there are few rigorous crystallization results in two and more dimensions. In this talk I will show how results by the Hungarian geometer László Fejes Tóth can be used to prove that an idealised block copolymer energy is minimised by the triangular lattice. I will also discuss a numerical method for a broader class of optimal location problems and some conjectures about minimisers in three dimensions. This is joint work with Mark Peletier, Steven Roper and Florian Theil. 

A number of phylogenetic algorithms proceed by searching the space of all possible phylogenetic (leaf labeled) trees on a given set of taxa, using topological rearrangements and some optimality criterion. Recently, such an approach, called BSPR, has been applied to the balanced minimum evolution principle. Several computer studies have demonstrated the accuracy of BSPR in reconstructing the correct tree. It has been conjectured that BSPR is consistent, that is, when applied to an input distance that is a tree-metric, it will always converge to the (unique) tree corresponding to that metric. Here we prove that this is the case. Moreover, we show that even if the input distance matrix contains small errors relative to the tree-metric, then the BSPR algorithm will still return the corresponding tree.