Quantum Gravity, Black Holes and the Big Bang are topics at the frontier of current theoretical physics research. They are related to challenging conceptual questions, but also involve very advanced mathematics. The main difficulty is the fact that all these phenomena require to take into account both quantum theory and Einstein's general relativity. In this talk I will give a pedagogical overview of current mathematical physics research into these questions.

Early aeronautical research in Britain was advanced by a decision to allow a number of the nation’s finest mathematicians to train as pilots and conduct airborne experiments using full-scale aircraft. Given that many would subsequently perish in flying accidents, what justified the risk?

Statistical physics is a fertile source of high-dimensional partial differential equations. We shall survey recent developments concerning a system of nonlinear partial differential equations, which involves the Navier--Stokes system coupled with a high-dimensional parabolic Fokker--Planck equation describing the motion of polymer molecules in a viscous fluid occupying a bounded spatial domain. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely or infinitely extensible nonlinear elastic spring potentials, and has been the subject of active research over the past decade. We shall review recent results concerning the existence of large-data global weak solutions to this high-dimensional system. We shall also highlight a number of nontrivial open questions concerning the mathematical analysis, approximation and numerical analysis of high-dimensional Navier--Stokes--Fokker--Planck systems.

A numerical library is a consistent, documented collection of quality software. The first published program was for computing the Bernoulli numbers, on Charles Babbage's analytical engine, by Ada Lovelace in 1842. The first published collection was in 1954 by Wilkes, Wheeler and Gill, primarily for the EDSAC computer at Cambridge University. This talk is essentially about the history of the development of quality numerical software, particularly numerical linear algebra software which has had a big influence on development in other areas. We shall look at the early influences and challenges and what I believe should be the attributes of quality software.

Predicting the outcomes of sports matches, based on information available in advance of the match, is of interest to a variety of parties – players, coaches, fans, bookmakers and gamblers. Some previous approaches have used computer simulations and/or stochastic models based on player statistics to predict such outcomes. In this paper, we instead investigate the use of regression models, using player or team rankings, previous performance in the same tournament, continent of origin and of the tournament, in predicting such outcomes of matches in the major “Grand Slam” tennis tournaments and international volleyball tournaments. We compare the use of “official” rankings of teams and players with ranking produced by a variant of a “page ranking” algorithm for internet web pages in these models, and study the success rate of each in predicting the results of matches not used for training the parameters of the models. We compare our results with those based on predicting that the higher ranked player/team, or the bookmaker’s favourite, would always win. Our models show a modest advantage over both simpler schemes. We then apply our models to both simple (fixed stake) and Kelly “odds overlay” betting strategies, placing virtual bets on each match. Our models were trained on “Grand Slam” results up to and including Wimbledon 2014 and the Volleyball 2014, and tested on the 2014 US Tennis Open and the 2014 Volleyball World Championships, neither of which was used in training the models. We find that in most cases, a Kelly betting strategy based on our model performs better than simpler approaches, and on average gives a positive return (unlike some of the other strategies).

Many meteorites are found on the surface of Antarctica. This is because Antarctica contains small regions called meteorite stranding zones where the upward flow of ice combines with a high level of ablation (ice loss) at the surface to enable large numbers of meteorites to be englacially transported to the surface.

In fact, up until December 2015, 34 927 meteorites had been recovered from the surface of Antarctica. This accounted for 66.3% of the world’s total number of collected specimens. Interestingly, of all the meteorites found in Antarctica only 0.7% of these were iron or stony iron meteorites. To put this into perspective 6.9% of finds throughout the rest of the word were iron based.

The hypothesis is that, due to their higher thermal conductivity, iron and stony iron meteorites heat up enough during the summer months to melt back down into the ice and offset the total annual upward transport of the ice.

To defend this hypothesis a simple 1D mathematical model of the movement of a meteorite through the ice will be presented. The use of the quasi-static heat equation will be defended and the attenuation of solar radiation within the ice will be discussed. I will then talk about the intriguing lack of meteorite finds in Greenland.

Synaptic vesicles are storage units for neurotransmitters that are released at the synapse. This release depends on an ion-induced Ca 2+ voltage gradient which regulates the rate of release of the neurotransmitters (the neuron fires a signal). If the quantity of released neurotransmitters is too small we get unusual nerve signals. Understanding the transport of synaptic vesicles is thus important in designing treatment for disorders such as ALS or Parkin- son’s disease. A particularly puzzling question in the field is how the high frequency of neuron firing can be achieved when the synaptic vesicles must traverse a very crowded environment to reach the cell membrane and release their cargo. In this talk I will introduce a mathematical model for the life cycle of synap- tic vesicles and discuss the role of the crowded environments on large-scale transport. The crowding effects can be reconciled with experimental obser- vations by introducing a zone of active transport or facilitated diffusion.

Inkjet printing based manufacturing of high-resolution p-OLED displays involves simultaneous printing of large arrays of pixels in close proximity. This requires the ability to control the spread of the deposited liquid within the desired pixel shape/region. It is usually achieved by patterning the substrates using a photo-lithography method. These patterns (referred to as pixels) are in the form of micron-sized topographical features or variations in the wettability of the substrate or a combination of both. Here we investigate how liquid deposited by sequential printing of partially overlapping droplets can be restrained within a stadium shaped pixel. We present a novel experimental system that can be used to study the dynamics of a single/multiple microdroplets deposited over patterned substrates. On substrates with only topographic variations, we find that the sloping side wall of the pixel can either locally enhance or hinder spreading depending on whether the topography gradient ahead of the contact line is positive or negative. Locally enhanced spreading occurs via the formation of thin pointed rivulets along the side walls of the pixel through a mechanism similar to capillary rise in sharp corners. However, the efficacy of sloped side-walls in restraining the spreading of deposited liquid in the desired region is critically dependent on the accurate positioning of droplets in a pixel. We demonstrate that this crucial dependence on the positioning of droplets can be evaded on the substrates with both topographical and wettability patterns.

Building on the experimental observations we demonstrate that a thin-film model combined with an experimentally measured spreading law, which relates the speed of the contact line to the contact angle, provides excellent predictions of the evolving liquid morphologies. We also show that the spreading can be adequately described by a Cox–Voinov law for the majority of the evolution. The model does not include viscous effects and hence, the timescales for the spreading of deposited liquid are not captured. Nonetheless, this simple model can be used very effectively to predict the areas covered by the liquid in a pixel and may serve as a useful design tool for systems that require precise control of liquid on substrates.

This talk will present a method to find the electromagnetic field and the Poynting vector resulting from a time-harmonic electromagnetic plane wave with no skew incidence and arbitrary polarization incident on an infinite perfect electric conducting wedge. Wedge diffraction is a widely explored area with numerous applications in acoustics, RADAR and ice crystal diffraction.

The aim is to reduce the electromagnetic problem to a scalar problem and find out how the polarization of this incident electromagnetic wave impacts the solution to diffraction by perfectly conducting wedges. I will use the third dimensional invariance of the scatterer and the boundary conditions to rewrite the electromagnetic field, governed by the source free Maxwell's equations, in terms of two uncoupled scalar potentials. These potentials depend on the polarization angle and can be shown to respectively solve the sound-soft and sound-hard scalar wedge problems. They can therefore be found using a variety of techniques including the Sommerfeld-Malyuzhinets technique and the Wiener-Hopf technique.

After this, I will compare multiple methods of evaluation or approximation of the electromagnetic field and the time-averaged Poynting vector for high frequency and plot the components of both quantities for different values of the polarization angle to determine its impact.

Uncertainty quantification (UQ) is rapidly gaining traction in the physical modelling and engineering communities and stochastic Galerkin finite element methods (SGFEMs) are now commonplace when approximating solutions to PDEs with random or parameter-dependent inputs. However, due to the tensor product structure of the approximation space, it is well known that SGFEMs quickly exhaust desktop computer memory. One technique to reduce the dimension of the approximation space is to initially construct a low-dimensional space, and use a posteriori error estimators to drive its incremental enrichment adaptively. For the enrichment strategy to be effective, the estimators must be accurate. We begin our talk by reviewing a classical a posteriori error estimation technique for weak problems. Following [A. Bespalov, C.E. Powell, and D. Silvester, Energy norm a posteriori error estimation for parametric operator equations, SIAM J. Sci. Comput., 36(2), 2014] we demonstrate that the weak parametric reformulation of the stochastic diffusion problem elegantly fits the classical framework. The effectivity of that estimator depends on a (deterministic) finite element space of our choosing. We investigate various spaces with the aim of designing an accurate error estimator (i.e., with effectivity indices close to one). For a model stochastic diffusion problem we demonstrate that very accurate and cheap-to-compute estimators are achievable.

We will introduce three mathematical approaches to investigate a number of problems of acoustic scattering by obstacles with simple geometries. Firstly, the simple method of separation of variables will be used to solve the problem of scattering by a cylinder subject to either a plane-wave or a point-source incidence. The half-plane problem will then be tackled with the more sophisticated Wiener--Hopf technique. Finally, we will give an outline of a probably less well known procedure used for the representation of the far-field of a solution: the embedding formula.

In this talk we consider inverse uncertainty quantification for the laser flash experiment. We assume the change in temperature of a material is modelled by the transient heat equation, with the diffusion coefficient considered unknown. We take the Bayesian approach, utilising Markov chain Monte Carlo (MCMC) methods to sample from the posterior distribution of the unknowns given observations of the temperature made during experimentation.

MCMC algorithms require many calculations of the posterior density. Each evaluation of the posterior density requires an evaluation of the forward model, which in this setting is the numerical solution of a time--dependent PDE. This requirement results in a computationally intensive routine, often taking days to achieve a desired Monte Carlo error in the quantities of interest. Although computationally intensive, such a routine is preferable to cheaper optimisation approaches which provide no quantification of the uncertainty, only a fitted best value.

We analyse the validity of using a surrogate model within an MCMC routine to reduce the computational cost. Specifically, we evaluate a single stochastic Galerkin finite element solution to the PDE in place of repeatedly computing deterministic finite element solutions within an MCMC routine. Investigations into how both the speed and accuracy of the approximation of the posterior are affected by this replacement are presented.