Smart Shrinkage Solutions - Fostering Resilient Cities in Inner Peripheries of Europe is a project that offers the best practice and most feasible solutions to the problem of urban shrinkage - a continuous population decline affecting more than 1,500 cities all over Europe. By learning from the experience of the cities that once were on the edge of an abyss but have bounced back to life, by sharing the key ingredients of their success across Europe and beyond, this project enables as many shrinking cities as possible to adapt, transform, and thrive in the face of continuously and often dramatically changing circumstances.

The peculiar behaviour of liquid and supercooled water has been baffling science for at least 236 years and is still seen as a major challenge facing chemistry today (Whitesides & Deutch, Nature 469, 21 (2011)). It was suggested that such strange behaviour might be caused by thermodynamic transitions, possibly even a second critical point. This second critical point would terminate a coexistence line between low- and high-density amorphous phases of water. Unfortunately, this second critical point (if it exists) and the associated polyamorphic liquid-liquid transition is difficult to study as it is thought to lie below the homogeneous nucleation temperature in a region known as "no man's land" (Angell, Science 319, 582 (2008)). In recent preliminary femtosecond optical Kerr-effect spectroscopy experiments, we have shown that water in concentrated eutectic solutions forms nanometre scale pools in which it retains many if not most of its bulk liquid characteristics. Most importantly, such solutions can be cooled to below 200 K without crystallisation (typically forming a glass at lower temperatures) allowing one to explore "no man's land" in detail for the first time. Preliminary experiments combining femtosecond spectroscopy with NMR diffusion measurements have shown that water in these pools undergoes a liquid-liquid transition as predicted for bulk water. Hence, it is proposed to use such nanopools as nanometre scale laboratories for the study of liquid and glassy water. A wide-ranging international collaboration has been set up to be able to study different critical aspects of the structure and dynamics of water. This includes cryogenic viscosity measurements, large dynamic-range (femtosecond to millisecond) optical Kerr-effect experiments, pulsed field gradient NMR, dielectric relaxation spectroscopy, terahertz time-domain spectroscopy, infrared pump-probe spectroscopy, and two-dimensional infrared spectroscopy. To ensure maximum impact of the experimental work, it is critical to have strong ties with experts in the theory and simulation of water and its thermodynamic behaviour. We have arranged collaboration with two international theory groups covering different aspects of the proposed work. Although the proposed research is relatively fundamental in nature, it will have impact as described in more detail elsewhere. The research addresses EPSRC priorities in nanoscience (supramolecular structures in liquids), energy (proton transport and liquid structuring in electrolytes for batteries and fuel cells), life sciences (the role of water in and on biomolecules), and the chemistry-chemical engineering interface (the role of the structuring of water in crystal nucleation). Our strong links with theory collaborators will ensure that fundamental insights will indeed propagate to the 'users' of such information. The close working relationship between the PI and CI has made Glasgow a centre of excellence in advanced femtosecond spectroscopy. This project exploits this expertise and international collaborations to immerse PDRAs and PGRSs in internationally leading research using state-of-the-art previously funded equipment.

The goal of this proposal is to ask for support from the Council towards the cost of a workshop on PDE which will be held in Oxford between June 28th and July 1st 2011.The mathematical analysis of PDE modelling materials presenting multiple scales have been an active area of research for more than 40 years. The study of the corresponding imaging, or reconstruction, problem is a more recent one. If the material parameters of the PDE present high contrast ratio, the solutions of the PDE become particularly challenging to analyse and to compute. Similar problems occur in time dependent equations in the frequency domain for high frequency. On the other hand, very high frequency regimes, or very contrasted materials, were considered first in imaging, as well-differentiated areas are, at first sight, simpler to locate by ad-hoc methods. Over the last decade the analysis of the inversion problem at moderate frequencies, the rigorous derivation of asymptotics at very high frequencies, and the regularity properties of solutions of elliptic PDE in very heterogeneous media have received a lot of attention.Part of the attention is due to the fact that these problems are particularly challenging. For another part, it is because of the numerous applications of these results in material sciences and in bio-medical imaging. Recently, emerging bio-medical imaging methods based on the observation of non-linear interactions of coupled physical phenomena (such as for example vibro-acoustography) have also become the subject of active research. Progresses on the mathematical understanding of the direct and inverse problems associated to these hybrid imaging methods are crucial to obtain enhanced imaging possibilities, beyond what is obtained by the integration of different imaging modalities taken separately. The focus of this workshop will be to stimulate collaborations between the participants, in the hope of achieving significant progress in (a) complete understanding of the direct problem with high contrast or high frequencies, (b) unified approaches to the inverse problem for both small and large contrast or frequencies, and (c) mathematical modelling of emerging experimental measurement methods. With this goal in mind, we wish to bring together senior experts and young researchers interested in the mathematical problems associated with imaging of multi-scale, or high contrast materials. All the mathematicians participating in the workshop are actively working on different aspects on these problems. Their expertise comprises heterogeneous random media, regularity theory for linear and non-linear PDE with very contrasted coefficients, mathematical invisibility (or cloaking), imaging and numerical reconstruction, numerical methods for high frequency elliptic problems, and emerging biomedical imaging methods. We have also invited an experimental physicist, whose recent work is devoted to new imaging methods for liquid crystals. The mathematical challenges associated with the mathematical formulation and understanding of these experiments and other hybrid measurement methods could be one of the applications of theoretical developments we hope this workshop will produce.

Corresponding to the three main objectives, there will be three parts of the project. A.) Study of exponentially small transitions in the Born-Oppenheimer model: Although the Born-Oppenheimer model is at the basis of much of quantum chemistry, there is no satisfactory theory describing the time development of exponentially small transitions between Born-Oppenheimer energy surfaces. Building on recent developments in space-adiabatic perturbation theory and on my work about the rigorous Darboux principle, I want to develop such a theory. A first step will be a thorough and rigorous understanding of low-dimensional situations, thus providing a major extension of the famous theory of universal time-adiabatic transitions by Michael Berry. Further effort will go into higher-dimensional models where probably only non-rigorous results will be possible. B.) Level-crossing dynamics in Quantum Chemistry: In quantum chemistry the systems under consideration are often too complicated to be treated rigorously, and it is unrealistic to achieve in A.) a theory which can fill all the needs that exist there. Moreover, today's powerful computers and advanced algorithms allow efficient numerical solutions of the time-dependent Schroedinger equation. On the other hand, there is often a lack of general paradigms allowing theoretical insight into processes involving level-crossing phenomena as occurring in femtosecond chemistry, and the method of space-adiabatic perturbation theory is as yet unknown in the field. In an interdisciplinary effort joint with Irene Burghardt (ENS Paris) I want to employ this method in order to develop theories for systems exhibiting conical or avoided crossings and probe these using the excellent numerical algorithms available. Here we will first concentrate on implementing the theory from A.) numerically and probing it against established algorithms in the field. Later we will move on to conical crossings in higher dimensions, using George Hagedorn's theory of conical intersections as well as space-adiabatic perturbation theory. Ultimately we aim at contributing to the recently very active topic of extremely high-dimensional conical intersections, where a scaling theory will likely be needed. In this part of the project I request a postdoc who will help me mainly on the numerical side of the research.C.) Rigorous results in asymptotic analysis: Sparked to some extent by the work of Michael Berry, exponential asymptotics has become a huge field, with a particularly strong school in the UK. There is now a wealth of knowledge about asymptotics beyond all orders, and there are many techniques for studying them. However, often for these methods the scope of their rigorous applicability is not entirely clear. A prominent exemple is the general Darboux Principle, which applies in various problems of great interest such as capillary waves or the selection of Saffman-Taylor fingers. In earlier work I introduced a tool which nails down the range of validity of this principle. I will apply this in order to treat problems like the above-mentioned which are well understood on a nonrigorous level, but also try to extend its use to new situations like multilevel nonadiabatic transitions. The results of this part of the project will form the basis for a book on rigorous exponential asymptotics.