When a fluid filled container is shaken vertically, one may observe waves on the surface of that container if the shaking is sufficiently strong. These waves arise out of a subharmonic instability: they have half the frequency of the shaking, and are called Faraday waves. In separate experiments, high-speed films of droplet impacts on static fluid baths show that the droplet does not always coalesce with the bath on impact, but may bounce a few times before coalescing. Combining these two experimental facts, about 10 years ago it was discovered that millimetric liquid droplets can bounce indefinitely when dropped on the surface of a bath of the same liquid in a shaken container. The phenomenon occurs below the shaking threshold for the Faraday instability. More surprisingly, the droplets can also spontaneously "walk" along the surface of the vibrating bath. These walkers then exhibit many features previously thought to be exclusive to quantum mechanics such as wave-particle duality, quantised energy states, single particle diffraction and tunnelling behaviour. The aim of this proposal is to explore the fluid mechanical aspects of this system. There are many unanswered challenging questions due to the complexity of the problem. Fluid mechanics questions include the understanding of non coalescing drop impact, and the behaviour of reflecting walkers and their pilot wave field at walls. We will also seek to understand how a purely classical mechanics system exhibits quantum mechanical-like behaviour, and probe the limits of this analogy. The proposed research involves a combination of analytical and numerical approaches as well as comparisons with experiments. We will partner with an MIT state-of-the-art fluid dynamics laboratory which will provide both experimental data and design validation experiments. The problems we plan to study are of general interest in fluid mechanics and in the theory of free boundary problems and dynamical systems. It is expected that the results will have broad applications, in particular to the understanding of the impact of drops and particles with fluids. Faraday instabilities are also the most reliable way of generating consistently sized droplets continuously. Because of this there are several possible microfluidics applications for this research, such as developing better devices for delivering inhaled drugs.

Random motions in random media have been intensively studied for over forty years and many interesting features of these models have been discovered. The aim is to understand the motion of a particle in a turbulent media. Most of the work has been focused on the case where the particle evolves in a static random environment, for which slow-downs and trapping phenomena have been proved. More recently, mathematicians and physicists have been interested in the case of dynamic random environments, where the media can fluctuate with time. Random walks on the exclusion process is probably the canonical model for the field. Much less is known on this model, but exciting conjectures and questions have been made. Some of the most challenging questions concern the possibility of super-diffusive regimes, and the existence of effective traps along the trajectory of the walk. In this project, we aim at, on one hand, adapt the techniques recently developped in one dimension to the multi-dimensional model and, on the other hand, understand the presence or absence of atypical behaviors for this model.

This proposal sits within a field of great scope, stretching from some of the most fundamental problems in physics, to current practical issues in engineering, to some of the most powerful modern techniques in topology and geometry. Although these topics are all very different, it has become apparent that many of the biggest future developments in each area will require overcoming key research challenges that are remarkably similar. It is these challenges that we will address in this proposed research. At the heart of each of the topics above lie Geometric Partial Differential Equations (PDE). Each of these equations could be perhaps a law of physics, or an equation modelling an industrial process, or more abstractly, a rule under which a geometric object can be processed in order to improve it. Smooth solutions to Geometric PDE have been extremely successful in applications to pure and applied problems, but the equations are generally nonlinear, and it is therefore typical that singularities will occur in solutions. The next generation of applications, with extensive potential impact, require us to transform our understanding of these singularities that develop. We must understand when and why they occur, their structure and stability, and how they encode what the PDE is doing. We must analyse to what extent they break the classical theory of smooth solutions, and what effects this has. These are the main challenges of this proposal, and we have compiled a team to address them with complementary expertise in singularity analysis and experience of applying geometric PDE across subjects such as Mathematical Relativity, Geometric Flows and Minimal Surfaces. In Mathematical Relativity, one sees singularities in solutions of the Einstein equations, first written down by Einstein in 1915 as the fundamental equations of the large-scale universe. Progress in the research challenges we propose will have potentially major impact in some of the most famous open problems in this field such as the Cosmic Censorship Conjectures, and the Black Hole Stability Problem. We also find singularities in the field of Geometric Flows, by which we mean the evolution equations of `parabolic' type that are currently being so successful in applications to geometry, topology and engineering, and in modelling phenomena in physics and biology. The most famous application in recent years has been the resolution of the Poincaré conjecture, which was named by the journal `Science' as the scientific `Breakthrough of the year, 2006,' but is considered by many to be the greatest achievement of mathematics in the past 100 years. The research challenges we propose are central to future applications of these equations, whether we are using them to classify manifolds with a certain curvature condition, or manipulate an image from a medical scanner. Intimately connected with these two subjects is the theory of Minimal Surfaces. These surfaces have been historically used to model soap films, but the general theory has developed into a powerful tool with applications to a wide range of subjects from black holes to topology. In this direction, we are particularly interested in applying progress on the research challenges of this proposal to unravel the connection between the existence of higher-index minimal surfaces and the singularities that occur in flows and variational problems that are designed to find them.

SAMBa aims to create a generation of interdisciplinary mathematicians at the interface of stochastics, numerical analysis, applied mathematics, data science and statistics, preparing them to work as research leaders in academia and in industry in the expanding world of big models and big data. This research spectrum includes rapidly developing areas of mathematical sciences such as machine learning, uncertainty quantification, compressed sensing, Bayesian networks and stochastic modelling. The research and training engagement also encompasses modern industrially facing mathematics, with a key strength of our CDT being meaningful and long term relationships with industrial, government and other non-academic partners. A substantial proportion of our doctoral research will continue to be developed collaboratively through these partnerships. The urgency and awareness of the UK's need for deep quantitative analytical talent with expert modelling skills has intensified since SAMBa's inception in 2014. Industry, government bodies and non-academic organisations at the forefront of technological innovation all want to achieve competitive advantage through the analysis of data of all levels of complexity. This need is as much of an issue outside of academia as it is for research and training capacity within academia and is reflected in our doctoral training approach. The sense of urgency is evidenced in recent government policy (cf. Government Office for Science report "Computational Modelling, Technological Futures, 2018"), through the EPSRC CDT priority areas of Mathematical and Computational Modelling and Statistics for the 21st century as well as through our own experience of growing SAMBa since 2014. We have had extensive collaboration with partners from a wide range of UK industrial sectors (e.g. agri-science, healthcare, advanced materials) and government bodies (e.g. NHS, National Physical Laboratory, Environment Agency and Office for National Statistics) and our portfolio is set to expand. SAMBa's approach to doctoral training, developed in conjunction with our industrial partners, will create future leaders both in academia and industry and consists of: - A broad-based first year developing mathematical expertise across the full range of Statistical Applied Mathematics, tailored to each incoming student. - Deep experience in academic-industrial collaboration through Integrative Think Tanks: bespoke problem-formulation workshops developed by SAMBa. - Research training in a department which produces world-leading research in Statistical Applied Mathematics. - Multiple cohort-enhanced training activities that maximise each student's talents and includes mentoring through cross-cohort integration. - Substantial international opportunities such as academic placements, overseas workshops and participation in jointly delivered ITTs. - The opportunity for co-supervision of research from industrial and non-maths academic supervisors, including student placements in industry. This proposal will initially fund over 60 scholarships, with the aim to further increase this number through additional funding from industrial and international partners. Based on the CDT's current track record from its inception in 2014 (creating 25 scholarships to add to an initial investment of 50), our target is to deliver 90 PhD students over the next five years. With 12 new staff positions committed to SAMBa-core areas since 2015, students in the CDT cohort will benefit from almost 60 Bath Mathematical Sciences academics available for lead supervisory roles, as well as over 50 relevant co-supervisors in other departments.