Spreading processes on geometric networks are often influenced by a network’s underlying spatial structure, and it is insightful to study the extent to which a spreading process follows that structure. In particular, considering a threshold contagion on a network whose nodes are embedded in a manifold and which has both 'geometric edges' that respect the geometry of the underlying manifold, as well as 'non-geometric edges' that are not constrained by the geometry of the underlying manifold, one can ask whether the contagion propagates as a wave front along the underlying geometry, or jumps via long non-geometric edges to remote areas of the network. 
Taylor et al. developed a methodology aimed at determining the spreading behaviour of threshold contagion models on such 'noisy geometric networks' [1]. This methodology is inspired by nonlinear dimensionality reduction and is centred around a so-called 'contagion map' from the network’s nodes to a point cloud in high dimensional space. The structure of this point cloud reflects the spreading behaviour of the contagion. We apply this methodology to a family of noisy-geometric networks that can be construed as being embedded in a torus, and are able to identify a region in the parameter space where the contagion propagates predominantly via wave front propagation. This consolidates contagion map as both a tool for investigating spreading behaviour on spatial network, as well as a manifold learning technique. 
[1] D. Taylor, F. Klimm, H. A. Harrington, M. Kramar, K. Mischaikow, M. A. Porter, and P. J. Mucha. Topological data analysis of contagion maps for examining spreading processes on networks. Nature Communications, 6(7723) (2015)

We will discuss what it means to study the homology of a group via the construction of the classifying space. We will look at some examples of this construction and some of its main properties. We then use this to define and study the homology of the mapping class group of oriented surfaces, focusing on the approach used by Harer to prove his Homology Stability Theorem.

When Gromov defined non-positively curved cube complexes no one knew what they would be useful for.
Decades latex they played a key role in the resolution of the Virtual Haken conjecture.
In one of the early forays into experimenting with cube complexes, Aitchison, Matsumoto, and Rubinstein produced some nice results about certain "cubed" manifolds, that in retrospect look very prescient.
I will define non-positively curved cube complexes, what it means for a 3-manifold to be cubed, and discuss what all this Haken business is about.
 

A graph of groups decomposition is a way of splitting a group into smaller and hopefully simpler groups. A natural thing to try and do is to keep splitting until you can't split anymore, and then argue that this decomposition is unique. This is the idea behind JSJ decompositions, although, as we shall see, the strength of the uniqueness statement for such a decomposition varies depending on the class of groups that we restrict our edge groups to

Machine learning for scientific applications faces the challenge of limited data. To reduce overfitting, we propose a framework to leverage as much as possible a priori-known physics for a problem. Our approach embeds a deep neural network in a partial differential equation (PDE) system, where the pre-specified terms in the PDE capture the known physics and the neural network will learn to describe the unknown physics. The neural network is estimated from experimental and/or high-fidelity numerical datasets. We call this approach a “deep learning PDE model” (DPM). Once trained, the DPM can be used to make out-of-sample predictions for new physical coefficients, geometries, and boundary conditions. We implement our approach on a classic problem of the Navier-Stokes equation for turbulent flows. The numerical solution of the Navier-Stokes equation (with turbulence) is computationally expensive and requires a supercomputer. We show that our approach can estimate a (computationally fast) DPM for the filtered velocity of the Navier-Stokes equations. 

Many complex systems can be represented as networks. However, it is often not possible or even desirable to observe the entire network structure. For example, in social networks, it is often difficult to obtain samples of large networks due to commercial sensitivity or privacy concerns relating to the data. However, it may be possible to provide a coarse grained picture of the graph given knowledge of the distribution of different demographics (e.g age, income, location, etc…) in a population and their propensities for forming ties between each other.

I will explore the degree to which it is possible to influence Ising systems, which are commonly used to model social influence, on unobserved graphs. Using both synthetic networks (stochastic blockmodels) and case studies of real world social networks, I will demonstrate how simple models which rely only on a coarse grained description of the system or knowledge of only the underlying external fields can perform comparably to more expensive optimization algorithms.

The talk will begin with an introduction to the science of what determines the behaviour of a liquid on a on a surface and giving an overview of some of the different theories that can be used to describe the shape and structure of the liquid in the drop. These include microscopic density functional theory (DFT), which describes the liquid structure on the scale of the individual liquid molecules, and mesoscopic thin film equation (PDE) and kinetic Monte-Carlo models. A DFT based method for calculating the binding potential ?(h) for a film of liquid on a solid surface, where h is the thickness of the liquid film, will be presented. The form of ?(h) determines whether or not the liquid wets the surface. Calculating drop profiles using both DFT and also from inputting ?(h) into the mesoscopic theory and comparing quantities such as the contact angle and the shape of the drops, we find good agreement between the two methods, validating the coarse-graining. The talk will conclude with a discussion of some recent work on modelling evaporating drops with applications to inkjet printing.

Dominic Vella will talk about writing grants, Anna Seigal will talk about writing research fellow applications and Jason Lotay will talk about his experience and tips for applying for faculty positions.