This is a first grant proposal for a collaboration project between the University of Liverpool (UL) and Max-Planck-Institute for Extraterrestrial Physics (MPE), the collaboration agreement is attached. The project aim is to use a new form of matter (complex plasma) to model phonons and solitons in solids and liquids in a unique and novel way. Complex plasmas is a highly interdisciplinary topic, which links many physics disciplines: plasma physics, solid state physics, acoustics, optics, nonlinear science, physics of electronic devices, gas and fluid dynamics, thermodynamics, and nanoscience. Complex plasmas is a new field of plasma physics, it allows to observe solid and liquid systems at the kinetic level (with motion of every ``molecule'' or grain resolved). This makes it possible to study linear and nonlinear wave phenomena (which we propose here) on the intermolecular length scales. Such a study in real solids and liquids would require following individual molecules in real time, which is almost impossible at the current level of technology. Waves and solitons with short wavelengths and their interactions are important for practical applications such as optical telecommunications and acoustic imaging. Their understanding in a model system - complex plasma can also lead to novel applications. The fundamental questions that can be answered using waves in complex plasmas include scattering of phonons on defects, nonlinear phonon-phonon interaction, mean free path of short-wavelength phonons (which affect thermal properties of solids and liquids), acoustic and thermal properties of nanostructures, nanoclusters, alloys, and semiconductors. Short-wavelength phonons also yield information about inter-grain forces and potentials. They are equivalent to terahertz phonons in solids which are being actively studied due to possible applications in sasers (Sound Amplification by Stimulated Emission of Radiation), generation of coherent phonon beams, and scanning acoustic microscopy (SAM). SAM has resolution comparable to soft X-ray imaging, but it is significantly less destructive to the material. Solitons are nonlinear localized waves which have a remarkable property of propagating without changing their shape and without destructive interaction with other solitons. This makes them very useful in optical telecommunications. Short-wavelength solitons are used in holey fibers to increase the baud rate of the transmission line. Due to energy dissipation the amplitude of the soliton decreases, while the width increases. Fine tuning the dispersion and nonlinearity of the medium, it is possibble to minimize the spreading of the soliton. As the soliton width decreases and becomes comparable with the intermolecular distances, it becomes more important to know how the medium anisotropy, scattering on defects, and interactions with other solitons affect propagation.We are planning to answer the long standing questions about the phase relation between the components in waves with mixed (longitudinal and transverse) polarisation and about the reduction of the phonon mean free path compare to the damping length of a monochromatic wave, possibly due to phonon interaction with other phonons and/or with defects.The proposed research gives us an opportunity to study waves and solitons at intermolecular scales. This is an extremely timely subject because of the growing interest for short wavelength wave phenomena and their direct applications in optical telecommunications, scanning acoustic microscopy, coherent phonon beams, etc.

Original application: 'The project's core deliverables are a Compendium, a Database and a Final Report by fourteen internationally distinguished scholars on the legal responses to Covid-19 in 80 countries across all regions of the world. The Compendium comprises 80 national reports written by local legal experts on the relevant country's response to Covid-19, covering: (1) the constitutional/legal framework; (2) the functioning of institutions (e.g. legislatures, courts); (3) the core public health measures adopted; (4) the social and economic measures adopted; and (5) key legal measures in respect of civil liberties and vulnerable groups. The Database collates determinate and quantifiable data on these themes, allowing users to conduct comprehensive cross-national comparisons and correlations with other known socio-economic, political and health data. The Final Report will comprise: 1. an analytical overview of the data, identifying response trends and correlations to major socio-economic and health indicators; and 2. an in-depth critical analysis of various thematic areas (e.g. privacy, civil liberties, migration), proposing best and worst practices in relation to different themes as well as overall state performance. The deliverables provide critical comparative data for the assessment of the UK's response to Covid-19 as well as for future pandemic preparedness, in general and with particular reference to several topics and questions identified as critical by UKRI: economic, gender and race inequalities; security and justice; national recovery and transformation; contact tracing; and national security and foreign policy. The project's dissemination plans include a clear and viable impact pathway into decision-making in the UK Parliament as well as in Whitehall.' From the www.lexatlas-c19.org website, more media friendly: "The Lex-Atlas: Covid-19 (LAC19) project was launched in the fall 2020 and will provide a scholarly analysis of national legal responses to Covid-19 around the world. Updated across 2021, it will be published open-access by Oxford University Press. It is the product of a vast collaboration of legal experts from across the world, led by University College London, King's College London, the Max Planck Institute of Comparative Public Law and generously supported by the UK's Arts and Humanities Research Council. The project is motivated by the need for a comprehensive overview of national legal responses to Covid-19. The pandemic has many facets, and national responses have varied considerably. Quite apart from epidemiological performance, countries have employed emergency powers differently, have had different kinds of institutional disruption, diverged in public health measures, and have had variable social policy coverage and responses to the human rights needs of vulnerable groups. A scholarly overview of these legal responses is required both to assess past political choices and to prepare for future pandemics. Cataloguing them in detail will also be an important contribution to the history of the pandemic. However, the complexity and fluid nature of the subject-matter essentially requires an unconventional scholarly approach. To make the international comparisons valuable, it requires a high degree of coordination between distinguished national legal experts, a large editorial team applying a consistent methodology, and the capacity to change national portraits as the law and policy shifts in line with the evolution of the pandemic. The project seeks to meet this need through a world-wide collaboration between legal scholars. The project's core deliverables include a Compendium of Country Reports, a Database, and a Final Report covering best and worst practices in the views of the project's Editorial Committee. All deliverables will be open-access and data will be held open-source. The project portal and further details are available at www.lexatlas-c19.org."

A huge number of important and challenging applications in operational research are governed by optimization problems. One crucial class of these problems, which has significant applicability to real-world processes, is that of partial differential equation (PDE)-constrained optimization, where an optimization problem is solved with PDEs acting as constraints. To provide one illustration, such formulations arise widely in image processing applications: this produces a crucial link to scientific and technological challenges from far-and-wide, for example determining the health of complex human organs such as the brain, exploring underground geological structures, and enabling Google cars to function without a human driver by assessing traffic situations. The possibilities offered by PDE-constrained optimization problems are immense, and consequently they have recently attracted tremendous interest from researchers in mathematics, as well as applied scientists more widely. These formulations may also be used to describe processes in fields as wide-ranging as fluid dynamics, chemical and biological mechanisms, other image processing problems such as medical imaging, weather forecasting, problems in financial markets and option pricing, electromagnetic inverse problems, and many other applications of importance. The study of these problems is therefore a cutting-edge research area, and one which can forge a huge advance in the fields of operational research and optimization. There has been much theoretical work undertaken on these problems, however the construction of strategies for solving these optimization problems numerically is a relatively recent development. In this project I wish to build fast and effective solvers for the matrix systems involved (these systems contain all of the equations which arise from the problem). The solvers are coupled with the development of a powerful 'preconditioner' (the idea of which is to approximate the corresponding matrix accurately in some sense, but in a way that is cheap to apply on a computer). Carrying this out is a highly non-trivial challenge for many reasons, specifically that it is often infeasible to store the matrix in its entirety at any one time, it is very difficult to build an approximation that captures the properties of the matrix in an effective way and is also cheap to apply, it is frequently necessary to build solvers which are parallelizable (meaning that computations may be carried out on many different computers at one time), and one is often required to carry out the expensive process of re-computing many different matrices. The aim of this project is to build powerful solvers, which counteract the above issues, for PDE-constrained optimization problems of significant real-world and industrial value. I will consider four specific applications: optimal control problems arising from medical imaging applications, PDE-constrained optimization formulations of image processing problems, models for the optimal control of fluid flow, and control problems arising in chemical and biological processes. I will consider problem statements that have the maximum practical potential, and generate viable, fast and effective solution strategies for these problems.

An ancient and essential development for life on Earth has been the evolution of a thin film of lipids that surround and form each cell. These membranes provide a barrier to water and hydrophilic solutes. They serve to isolate biological reactions from the outside, and offer the potential to separate charge and to communicate and combine with one another to form complex structures. More complicated cells also contain internal membrane structures that provide a further division for chemical reactions and generation of electric potentials. These events facilitated the ability to harness energy and to develop and maintain the complex structures and biochemistry of the cell. The necessary exchange of materials across lipid membranes between the outside and different compartments gives rise to a transport problem for small and large molecules alike. Proteins are large polymers of amino acids made according to the genetic code of each respective gene in the cell cytosol. In order to perform their specific roles many of them need to be specifically targeted and delivered to alternative locations. This requires that they pass either across or into a specific membrane. This proposal aims to learn more about this important process using the bacterial cell membrane as a model system. The apparatus responsible for protein movement across membranes has been purified. The interaction of this protein channel and its polypeptide substrate will be studies using a variety of biochemical and biophysical approaches that we have at our disposal. New findings in this area will have implications in the understanding of protein translocation. Moreover, they will also help us understand numerous other important problems in biology that involve the interaction of protein complexes with polypeptides.

The role of condensed matter theory is to gain a deep understanding of phenomena observed in nature, either in the laboratory or in the natural world, and to use these insights to predict new behaviours and inspire novel directions for experimental design and investigation. To develop this approach, as theorists, we construct models of physical and chemical processes that are often inspired by experimental discoveries. For some applications, these models may be derived from a basic knowledge of the fundamental interactions of constituent particles; in other cases, they may be abstractions inspired by experimental phenomenology. We then make use of computational and analytical approaches to explore and refine these models to capture and explain known experimental properties and, crucially, to make testable predictions about new phenomena. By placing emphasis on "emergent" or collective behaviours of complex systems, the models we study can often provide insight into disparate fields of research, spanning across areas of physics, chemistry, material science and biology. As a result, our theoretical research activities are often interdisciplinary in character, contributing both to fundamental knowledge creation and providing practical applications of modelling to new and existing technologies. A key strength of our research is its reliance on concepts and methodologies that can be shared and translated across seemingly disparate subject areas. To benefit from this approach, we seek support in the form of Critical Mass funding that will enable us to recruit post-doctoral researchers who can profit from engagement with more than one investigator, enabling them to strengthen and create new collaborations, open new areas of research, and advance their own ideas. The proposed programme of research is separated broadly across three different themes which, together, are united by the common theme of dynamical phenomena. By calculating and interpreting the characteristics of electrons in materials, we can provide deep explanations for common phenomena like the flow of electricity or heat, which in turn suggest new pathways for technological innovation. The research proposed in Theme 1 will advance our understanding, and our ability to predict many aspects of materials behaviour, ranging from superconductivity to magnetism. In Theme 2, we propose an interconnected programme of research on dynamical aspects of quantum many-particle systems, addressing: finite-temperature experimental signatures of quantum spin liquids in novel magnetic materials; the nature of topologically protected excitations in open quantum systems; and dynamical aspects of quantum circuits and their links to quantum error correction. Finally, in theme 3, we will use computational and analytical approaches to study the dynamics of classical systems far from equilibrium, focusing on the study of glass-like phenomena in constrained systems, finding applications in both particulate and living matter.