Oscillations are everywhere, ranging from perfectly ordered periodic and quasiperiodic to completely disordered, irregular ones, often described by probabilistic laws. Turbulence, climate changes, neuron spiking, heart electrical activity / all of these are examples of processes where irregular oscillations are possible and play a prominent role.The irregularity of oscillations can have two different origins: deterministic and stochastic.In the former case, although the dynamics of the oscillating system is defined by deterministic laws, the oscillations themselves are very sensitive to the initial state of the system: even a very small change in this initial state can lead to a substantial difference in the behaviour. This kind of oscillations is usually called deterministic chaos. Another type of irregular oscillations occurs when the dynamics of the system is defined by random fluctuations existing within, or applied to, the system. Notably, in spite of their randomness, the noise-induced oscillations in some systems can look quite regular and resemble very much the deterministic ones. One of the brightest examples is a sensory neuron which demonstrates no oscillations unless the signal on its input exceeds a certain level, after which the neuron generates one electrical pulse whose shape and duration are defined deterministically and almost do not depend on the input. The usual signal coming from an environment is in fact a random signal, that being applied to the neuron can generate a sequence of pulses looking quite coherent. Amazingly, irregular oscillations of this kind are widely spread in nature and technology. Besides the neurons and neuron networks, such oscillations can arise in semiconductor nanostructures, chemical reactions, some engineering mechanisms like drill string and many others. It is obvious that our ability to control irregular oscillations for example by making them more predictable is hugely beneficial to industry, technology, medicine, etc. Control means that by imposing some, preferrably small, forcing or feedback on the system, one is able to change the amplitude, timescale or regularity of oscillations, or even to cease them altogether. In the last decade a good progress was made in the control of deterministic chaos. The advanced nonlinear control methods exploit the fact that deterministic trajectories with the desired timescales already exist in the system, but are unstable and thus invisible in experiment, and the control tools just stabilize them. However, the systems where oscillations occur only due to noise have no such trajectories and no deterministic timescales. All timescales or orbits can be introduced only in the statistical sense. The control of oscillations that are purely random has never (or rarely, depending on what one means by control) been addressed previously.The main aim of the current research is to develop a general effective method for the control of oscillations induced merely by external random fluctuations that would be feasible as applied to real-life problems. As a control tool delayed feedback will be considered, which looks the most promising from the viewpoint of simplicity and efficiency. The main objectives are: (i) To develop qualitative theories for the delayed feedback as applied to minimal models that describe a large class of nonlinear systems in which noise can induce oscillations. (ii) To establish if the method is applicable to more realistic models like neuron-like networks. (iii) To verify if the delayed feedback can work in a real experiment on control of heart rate and of its variability in experiments with healthy human volunteers. The work will be carried out in Loughborough University in collaboration with Technical University of Berlin and the Department of Cardiovascular Sciences of Leicester University.

The context of the proposal mainly concerns singular stochastic PDEs and related statistical physics models. By saying singular, we mean that the solution (or some of its derivatives) has wild oscillations with a frequency and magnitude blowing up to infinity at small scales. The singularities in the solutions to stochastic PDEs are typically almost everywhere. As a consequence, nonlinear operations of the solutions may not make sense as they take these high frequency oscillations into quantities that are typically infinity. Thus, the correct interpretation of the solutions to these equations usually requires renormalisation. In the past three years, there have been major advances in the development of solution theories to a number of important singular SPDEs, including the three dimensional stochastic quantisation equation, the KPZ equation and the parabolic Anderson model in two and three dimensions. These equations are widely believed to be the universal models for the large scale behaviours of many systems in statistical mechanics. The successful construction of the solutions opens a way to study in detail these equations as well as the natural phenomena they represent. In this proposal, we aim to deepen the understanding of the quantitative behaviour of the solutions to these equations, and rigorously prove the universality phenomena for their related statistical physics models. We will also investigate how certain perturbations of the system (for example, asymmetry in phase coexistence models) can force its large scale behaviour to deviate from the expected universal limit.

The context of the proposal mainly concerns singular stochastic PDEs and related statistical physics models. By saying singular, we mean that the solution (or some of its derivatives) has wild oscillations with a frequency and magnitude blowing up to infinity at small scales. The singularities in the solutions to stochastic PDEs are typically almost everywhere. As a consequence, nonlinear operations of the solutions may not make sense as they take these high frequency oscillations into quantities that are typically infinity. Thus, the correct interpretation of the solutions to these equations usually requires renormalisation. In the past three years, there have been major advances in the development of solution theories to a number of important singular SPDEs, including the three dimensional stochastic quantisation equation, the KPZ equation and the parabolic Anderson model in two and three dimensions. These equations are widely believed to be the universal models for the large scale behaviours of many systems in statistical mechanics. The successful construction of the solutions opens a way to study in detail these equations as well as the natural phenomena they represent. In this proposal, we aim to deepen the understanding of the quantitative behaviour of the solutions to these equations, and rigorously prove the universality phenomena for their related statistical physics models. We will also investigate how certain perturbations of the system (for example, asymmetry in phase coexistence models) can force its large scale behaviour to deviate from the expected universal limit.

How do we understand whether epidemics will spread, or predict the likelihood of extreme meteorological events? How do we optimize nano-particles to make them efficient vehicles for targeted delivery of drugs in medical applications? Can we predict how a cell in a specific state will evolve in time, for example whether it will stay healthy or become diseased? How do we prevent over-heating in fast electronic devices, or understand how energy conversion in the next-generation of solar cells might work? Can insights into extreme events in the sciences be used to detect whether a network of financial institutions is close to a crash? These research challenges all relate to non-equilibrium systems. Such systems are typically irreversible, so that if a movie of the system was played backwards it would look very different. For equilibrium systems, on the other hand, a backwards movie would appear much like the original. This lack of an "arrow of time" makes equilibrium systems relatively easy to understand, and indeed most existing methods for predicting e.g. the behaviour of materials are based on the assumption of equilibrium. For non-equilibrium systems, on the other hand, we know much less. They can "age" towards an equilibrium state that is never reached, or exhibit extreme events. The latter are often the result of cascades of collective failure, as illustrated in phenomena ranging from stock market crashes to environmental disasters. As the examples in the first paragraph show, understanding, predicting and controlling non-equilibrium behaviour is an important challenge in many problems across the physical, mathematical, biological and environmental sciences. The starting point for the proposed interdisciplinary Centre for Doctoral Training is, therefore, that significant progress will require researchers that can exploit and strengthen such links between disciplines. They will need to be trained in how to analyse non-equilibrium systems theoretically, via mathematical models, how to study their behaviour via computer simulations, and how to extract information about them from possibly noisy or incomplete data. They will also need to be aware of the important non-equilibrium problems in different disciplines, to see connections, transfer methods and concepts from one application area to another, and develop new approaches. The Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES) will provide the training that such a new generation of researchers will need. In a substantial cohort of at least 10 PhD students per year it will make sure that ideas are exchanged systematically; research projects will be designed to build bridges between different disciplines employing similar methods, or to explore the connections between different approaches that are used to study non-equilibrium systems in the same area. Students will acquire transferable communication and presentation skills, and take part in outreach activities to increase the public understanding of non-equilibrium science. In `open questions sandpits', industry engagement events and dedicated careers events they will also obtain a solid understanding of the priorities of industrial partners working on non-equilibrium systems, and of attractive career paths outside of university. Overall, CANES researchers will emerge with a wide variety of skills that are highly sought after in academia and industry. CANES will also put UK research in non-equilibrium systems on the international map, helping the UK to compete against other countries like the U.S.A. where there is already a significant drive to strengthen research in this area.

Overview: We propose a Centre for Doctoral Training in Data Science. Data science is an emerging discipline that combines machine learning, databases, and other research areas in order to generate new knowledge from complex data. Interest in data science is exploding in industry and the public sector, both in the UK and internationally. Students from the Centre will be well prepared to work on tough problems involving large-scale unstructured and semistructured data, which are increasingly arising across a wide variety of application areas. Skills need: There is a significant industrial need for students who are well trained in data science. Skilled data scientists are in high demand. A report by McKinsey Global Institute cites a shortage of up to 190,000 qualified data scientists in the US; the situation in the UK is likely to be similar. A 2012 report in the Harvard Business Review concludes: "Indeed the shortage of data scientists is becoming a serious constraint in some sectors." A report on the Nature web site cited an astonishing 15,000% increase in job postings for data scientists in a single year, from 2011 to 2012. Many of our industrial partners (see letters of support) have expressed a pressing need to hire in data science. Training approach: We will train students using a rigorous and innovative four-year programme that is designed not only to train students in performing cutting-edge research but also to foster interdisciplinary interactions between students and to build students' practical expertise by interacting with a wide consortium of partners. The first year of the programme combines taught coursework and a sequence of small research projects. Taught coursework will include courses in machine learning, databases, and other research areas. Years 2-4 of the programme will consist primarily of an intensive PhD-level research project. The programme will provide students with breadth throughout the interdisciplinary scope of data science, depth in a specialist area, training in leadership and communication skills, and appreciation for practical issues in applied data science. All students will receive individual supervision from at least two members of Centre staff. The training programme will be especially characterized by opportunities for combining theory and practice, and for student-led and peer-to-peer learning.