Warwick EPSRC Symposium on Fluctuation-driven Phenomena and Large Deviations

The Warwick EPSRC mathematics symposium is organised annually by the University of Warwick with the support of the EPSRC for the benefit of the mathematical sciences community in the UK. It brings leading national and international experts together with UK researchers in a year-long programme of research activities focused on an emerging theme in the mathematical sciences. The proposed symposium for the 2015-16 academic year will concentrate on the theme of "Fluctuation-driven phenomena and large deviations". In very general terms, the symposium will constitute an interdisciplinary focus on understanding the consequences of the interplay between stochasticity and nonlinearity, a recurrent challenge in many areas of the mathematical sciences, engineering and industry.

Stochastic processes play a fundamental role in the mathematical sciences, both as tools for constructing models and as abstract mathematical structures in their own right. When nonlinear interactions between stochastic processes are introduced, however, the rigorous understanding of the resulting equations in terms of stochastic analysis becomes very challenging. Mean field theories are useful heuristics which are commonly employed outside of mathematics for dealing with this problem. Mean field theories in one way or another usually involve replacing random variables by their mean and assuming that fluctuations about the mean are approximately Gaussian distributed. In some cases, such models provide a good description of the original system and can be rigorously justified. In many cases they do not. Understanding the latter case, where mean-field models fail, is the central challenge of this symposium. We use "fluctuation driven phenomena" as a generic term to describe the kinds of effects which are observed when mean field theories fail.

The challenges stem from the fact that the rich phenomenology of deterministic nonlinear dynamics (singularities, nonlinear resonance, chaos and so forth) is reflected in the stochastic context by a variety of interesting and sometimes unintuitive behaviours: long range correlations, strongly non-Gaussian statistics, coherent structures, absorbing state phase transitions, heavy-tailed probability distributions and enhanced probabilities of large deviations. Such phenomena are found throughout mathematics, both pure and applied, the physical, biological and engineering sciences as well as presenting particular problems to industrialists and policymakers. Contemporary problems such as the forecasting of extreme weather events, the design of marine infrastructure to withstand so-called "rogue waves", quantifying the probability of fluctuation driven transitions or "tipping points" in the climate system or estimating the redundancy required to ensure that infrastructure systems are resilient to shocks all require a step change in our ability to model and predict such fluctuation-driven phenomena. The programme of research activities constituting this symposium will therefore range from the very theoretical to the very applied.

At the theoretical end we have random matrix theory which has recently emerged as a powerful tool for analysing the statistics of stochastic processes which are strongly non-Gaussian without the need to go via perturbative techniques developed in the physical sciences such as the renormalisation group. At the applied end we have questions of existential importance to the insurance industry such as how to cost the risk of extreme natural disasters and quantify their interaction with risks inherent in human-built systems. In between we have research on the connections between large deviation theory and nonequilibrium statistical mechanics, extreme events in the Earth sciences, randomness in the biological sciences and the latest numerical algorithms for computing rare events, a topic which has seen strong growth recent years.

The main expected impacts of the symposium are as follows:

- the research expertise of the UK mathematical sciences community in the area of fluctuation-driven phenomena will be strengthened and the impact of research carried out in the UK will be enhanced. This will be achieved primarily as a result of the high international profile of the workshop programme and by ensuring that the programme is well advertised throughout the community.

- the exchange of ideas between more theoretical branches of the mathematical sciences and scientists working on applications will be enhanced. This will be achieved by the fact that applications feature strongly in the workshop programme.

- UK companies and policy-makers working in risk management, resilience planning, disaster recovery and insurance will be more aware of the most recent advances in the theory of rare events, extreme fluctuations and strong correlations emerging from academia. Equally, academics will be more aware of the kinds of problems which are important outside of academia which may guide future research directions and foster new collaborations at the industry-academia interface. This will be achieved by the inclusion of the industry-facing workshop in the programme which will discuss these issues and provide valuable networking opportunities.

Looking forward on the 10 to 50 year timescale, the kind of cross-disciplinary synthesis between the mathematical sciences and the wider scientific community which will be facilitated by this symposium will have a large impact on the UK's ability to respond to the threats and opportunities associated with future environmental and socio-economic challenges, many of which will require the ability to model and quantify extreme events and/or correlated fluctuations in highly connected systems. The simultaneous occurrence of a record-breaking nationwide freeze in the United States with a record-breaking nationwide heatwave in Australia in January 2013 provides a stark illustration of the fact that the threat posed by climate change to the UK and the world does not come primarily from the mean changes but from changes in the statistics of extreme events. On the socio-technological side, as the world becomes ever more interconnected, the potential for an unexpected fluctuation in one part of a system to initiate a cascade process resulting in large scale failure becomes an increasing threat. The problem is already well-recognised in the context of infrastructure networks such as power and transport. The Fukushima accident has illustrated how the consequences of our inability to properly account for correlations between natural and man-made sources of risk can have catastrophic consequences. The threat of correlated failures is likely to become more profound in a future where "everything" becomes online. The "internet of things" envisages a future where all of our manufacturing, transport and communications infrastructure combine with our personal and domestic technology to form a network of unparalleled complexity. This network is likely to pose unprecedented problems in terms of design and control in addition to unprecedented opportunities for innovation. That mathematics will be central in responding to these changes is not in doubt. The question is whether the right mathematical tools have been developed and integrated into the toolkits of the right people. It is in this latter respect that this symposium can hope to have the most lasting impact.