Physical systems of interest to society are extremely complex. For example, the atmosphere and ocean of Earth, relevant for questions of global warming, long time climate estimates or prediction of extreme whether events, consists of a tremendeous number of interacting systems, each with many degrees of freedom. It is impossible to consider the system in its full complexity when trying to make predictions, in particular if the questions concern long-time prediction or rare events. Indeed, multiscale systems are ubiquitous in nature: the underlying processes relevant to many physical phenomena often happen on vastly different length- or time-scales. This opens up the possibility of coarse-graining and averaging, where fast, fluctuating degrees of freedom can be considered as effective noise on slow degrees of freedom. In particular, this helps to reduce, or coarse-grain, complex physical models into much simpler models that are tractable analytically or numerically to enable prediction and deeper understanding of the involved processes. Rare events, for example conformational changes of the relevant unknowns, are particularly interesting and rich in such a setup. In stochastic systems, unlikely fluctuations can push the system from its typical state into other, meta-stable configurations with often vastly different properties. Examples are chemical reactions, phase transitions, weather patterns, protein folding, or persistent structures in fluid flow. In such situations, large deviation theory gives precise and rigorous estimates of the probabilities and mechanisms of these conformational changes, by generalising the notion of free energy and entropy to arbitrary stochastic systems. Obtaining explicit large deviation principles in this multiscale setup is a big challenge, since the associated fluctuations stem from averaging of complex physical processes, and therefore are generally non-linear, non-Gaussian, or even non-Markovian. The computation of large deviation principles in such a setup is of high importance, as it would allow us to estimate transition probabilities on the effective, coarse-grained model, without the need to consider all (fast, unimportant) degrees of freedom, thus making computation feasible. The proposal concerns itself with the development of theory and numerical algorithms in the above situation, and to make available the developed techniques to applied sciences. The PI will apply these large deviation methods for multiscale systems and coarse-grained models to three concrete problems: (i) Metastability in atmospheric jets, where turbulent fluctuations facilitate the disappearance of planetary jets in atmospheric flow, (ii) magnetically confined fusion experiments, where conformational changes in the boundary layer in plasma reactors prevent efficient confinement, and (iii) fibre-optics communications, where random fluctuations in optical fibres lead to bit-flips in photonic communication. All theoretical research efforts will result in the development of algorithms or software implementations permitting the re-use by researchers in other fields that are concerned with rare events in multiscale systems.

The discovery that matter is made up of atoms ranks as one ofmankind's greatest achievements. Twenty first century science isdominated by a quest for the mastery (both in terms of control andunderstanding) of our environment at the atomic level.In biology, understanding life (preserving it, or even attempting tocreate it) revolves around large, complex, molecules -- RNA, DNA, andproteins.Global warming is dictated by the particular way atoms are arrangedto make small greenhouse gas molecules, carbon dioxide and so on.The drive for faster, more efficient, cheaper computer chips forcesnanotechnology upon us. As the transistors that make up themicroscopic circuits are packed ever closer together, electronicengineers must understand where the atoms are placed, or misplaced, inthe semiconducting and insulating materials.Astronomers are currently, daily, discovering new planets outside oursolar system, orbiting alien stars. The largest are the easiest tospot, and many are far larger than Jupiter. The more massive theplanet the higher pressures endured by the matter that makes up itsbulk. How can we hope to determine the structure of matter at theseconditions?The atomic theory of matter leads to quantum mechanics -- a mechanicsof the every small. In principle, to understand and predict thebehaviour of matter at the atomic scale simply requires the solutionof the quantum mechanical Schroedinger equations. This is a challengein itself, but in an approximate way it is now possible to quicklycompute the energies and properties of fairly large collections ofatoms. But is it possible to predict how those atoms will be arrangedin Nature - ex nihilo, from nothing but our understanding ofphysics?Some have referred to it as a scandal that the physical sciencescannot routinely predict the structure of even simple crystals -- butmost have assumed it to be a very difficult problem. A minimum energymust be found in a many dimensional space of all the possiblestructures. Those researchers brave enough to tackle this challengehave done so by reaching for complex algorithms -- such as geneticalgorithms, which appeal to evolution to breed ever betterstructures (with better taken to mean more stable). However, Ihave discovered to my surprise, and to others', that the very simplestalgorithm -- throw the collection of atoms into a box, and move theatoms downhill on the energy landscape -- is remarkably effectiveif it is repeated many times.This approach needs no prior knowledge of chemistry. Indeed thescientist is taught chemistry by its results -- this is critical ifthe method is to be used to predict the behaviour of matter underextreme conditions, where learned intuition will typically fail.I have used this approach, which I call random structure searching to predict the structure of crystals ex nihilo. My firstapplication of it has been to silane at very high pressures, and thestructure I predicted has recently been seen in experiments. Butprobably the most impressive application so far has been to predictingthe structure of hydrogen at the huge pressures found in the gas giantplanets, where it may be a room temperature superconductor.In the course of my fellowship I will extend this work to try toanticipate the structure of matter in the newly discovered exoplanets,to try to discover and design materials with extreme (and hopefully,extremely useful) properties, and to help pharmaceutical researchersunderstand the many forms that their drug molecules adopt when theycrystallise.

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.

Stochastic partial differential equations (SPDE) describe the behaviour of spatially extended systems under the influence of noise. They arise naturally in various fields of applications as diverse as data mining, mathematical finance, and population dynamics and genetics. The present proposal aims to study a class of stochastic partial differential equations from statistical mechanics. Many particle models exhibit a behaviour called phase transition, where the behaviour of the system changes drastically when one changes a given system parameter beyond a critical point. It is a very exciting question to understand the behaviour of such a system near a critical point. In such a regime one expects the dynamics to be governed by a non-linear SPDE. Analytically the understanding of these equations is very challenging, because of the interaction between the rough noise term and the non-linear evolution. But this is also what gives rise to interesting phenomena. In this proposal, we aim to deepen the understanding of these equations. On the one hand we will study the behaviour under the influence of a small noise term. Then we will establish that two different kinds of particle models can indeed be described by such an SPDE.

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.