Averting dangerous consequences of climate change and transitioning to societies that use our natural resources sustainably is one of the most existential challenges currently facing humanity. At the technological level, advanced energy materials are needed not only to sustain incremental advances in existing zero-carbon energy technologies, but also to address open technology challenges requiring disruptive breakthroughs. New, emerging classes of energy materials, such as perovskite semiconductors and organic/biologically inspired materials for solar energy harvesting and photovoltaics, or advanced electrode materials for batteries offer great opportunities for achieving higher performance, lower cost and better environmental sustainability than existing energy materials. However, many aspects of their operation remain poorly understood. This is related to their relatively disordered, non-single crystalline microstructures, with complex interfaces that are critical for device operation, and the presence of weakly, non-covalently bonded, functional groups and molecular units. This makes the materials mechanically soft and the dynamics of lattice vibrations has a strong effect on the charge carriers and electronic excitations. However, their performance is surprisingly tolerant to such static and dynamic disorder, which opens a wide space for materials exploration as we apparently do not always need structural perfection. This centre-to-centre collaboration brings together a team of energy materials researchers at the Universities of Cambridge and Oxford supported by the VETSOFT EPSRC programme grant with a world-leading group of researchers at Princeton University's Andlinger Centre for Energy and the Environment. Both centres have internationally leading, interdisciplinary teams with a broad spectrum of complementary techniques and scientific capabilities that can be applied and shared across traditional boundaries associated with different materials systems and/or applications. By not working in traditional silos, powerful synergies can be achieved. This is at the heart of the VETSOFT programme grant, which brings together researchers working in soft functional energy materials for diverse applications in photovoltaics, photocatalysis, thermal energy harvesting and energy storage. A similar philosophy also underpins Princeton's Andlinger Centre, which has available a largely complementary set of capabilities. The proposed centre-to-centre collaboration aims to achieve a deeper atomistic understanding and control of important physical processes in soft functional energy materials, in turn driving tangible enhancements in energy materials performance and new device concepts. We have identified three grand research challenges (RCs) for which there is a high added value from the collaboration between the two centres and for which complementary scientific capabilities and methodologies available at the two centres are needed. The centre-to-centre collaboration will allow us to tackle these in a more effective way than any of the participating groups could on their own. The first two RCs address scientific bottlenecks that are holding back the application of perovskite semiconductors in solar cells and of electrode materials for batteries: We will develop approaches for controlled doping of metal halide perovskite semiconductors and new battery anode materials based on niobium tungsten oxides capable of fast charging. The third one aims to achieve a deeper, fundamental understanding of energy transfer processes in biological energy harvesting. The proposed centre-to-centre collaboration will also provide a vehicle for encouraging other, exploratory research projects in advanced energy materials between groups at the two centres, that will lead to a sustained, effective partnership between the two centres outlasting the 4-year funding period of the proposed project.

Many of the most challenging conundrums currently being addressed by frontier scientific research in astrophysics involve interactions between exotic objects of colossal sizes and/or energies, typically resulting in instances of extraordinary energy release: - the electromagnetic fireworks accompanying black-hole mergers, which are now observable with the advent of gravitational- wave and multi-messenger astronomy; - galaxy formation in clusters; - accretion discs and jets, which are now serially observable by the Event Horizon radio-telescope network; - gamma-ray and fast radio bursts; ultra-high-energy cosmic rays; and many other occurrences. To model these phenomena, a key challenge is to have a detailed understanding of the dilute hot gas (known as `plasma') making up the astrophysical environments where these events occur. Unsurprisingly, this plasma is believed to behave very differently to the gases we all encounter in everyday life, on account of being millions of degrees hotter, and one sextillionth the density! While this state of matter has been studied by physicists for nearly a century - most famously, in the contexts of stars and nuclear fusion energy research - there remain a number of surprisingly fundamental uncertainties about its properties: for example, how do plasmas conduct heat, and what is their viscosity? However, recent technological advances in both our computing capabilities and high-energy laser facilities mean that we can now investigate the behaviour of plasmas as never before in the laboratory and on supercomputers. In this research project, I will be undertaking a systematic programme that will significantly advance our understanding of the fundamental properties of the type of plasma typically encountered in astrophysical environments (whose thermal energy exceeds their magnetic energy). More specifically, I will run numerical simulations with state-of-the-art codes to investigate several different characteristics: viscosity, thermal and electrical conductivity, and the spontaneous generation of charged particles with anomalously high energies. I am particularly interested in behaviours which depart markedly from conventional gases. I will then test theoretical frameworks developed in "laboratory astrophysics" experiments, which use lasers to realise extreme conditions on Earth with many similarities to relevant astrophysical environments. In addition to the astrophysical observations, I am also interested in leveraging anomalous properties of magnetised plasmas to aid inertial confinement fusion (ICF) efforts. In ICF schemes, a small capsule of deuterium-tritium fuel is ignited using laser beams; if the scheme is successful, the resulting nuclear fusion reactions produce much more energy than initially applied with the lasers. At present, successful ICF schemes have not yet been achieved; however, I believe that significant improvements to current attempts could be attained by considered use of applied magnetic fields.

Statistics plays a fundamental role in daily life, allowing costly medical screening, drug development, marketing campaigns or government regulation to be better targeted through improved understanding of the scientific or societal truths underpinning the data we observe. More and more frequently, the scientific truths we wish to learn correspond to a high dimensional parameter. This project considers covariance matrices and related quantities such as inverse covariance matrices, which are particularly important types of high dimensional parameter, arising in numerous statistical applications. When the dimensionality of the covariance matrix is larger than the number of available data points, structure (sparsity in some domain) must be assumed in order to obtain estimates that are well behaved statistically. This project explores new types of structure for covariance and inverse covariance matrix estimation. Some of these structures facilitate uncertainty statements about the true high dimensional parameter rather than simply providing a point estimate. They also allow different estimates to be aggregated without losing statistical accuracy.

Entanglement is one of the most profound concepts to emerge from quantum mechanics, and a phenomenon whose implementation in real materials requires exceptional control over state preparation, coherence, coupling and measurement. The collaborators have already begun to address these individual challenges using the complementary advantages of both electron and nuclear spin degrees of freedom in a diverse range of materials, with notable successes including the coherent storage of the electron spin state in the nuclear spin to achieve coherence times of several seconds, and the true entanglement of an electron and nuclear spin with high fidelity. In this proposal, we will bring together these individual components, exploiting the transient nature of the electron spin in systems such as optically-excited molecules and silicon-based devices, in order to mediate entanglement between multiple nuclear spins. In addition to providing a key component of emerging quantum technologies within the solid state, this will lead to a new understanding of the mechanisms and correlations behind decoherence of electron and nuclear spins states, under different environments and processes.The key idea in this proposal is to use the transient electron spins in certain materials and devices, not only to understand and overcome spin decoherence mechanisms, but also to mediate the entanglement of multiple nuclear spins. Through experiment, density functional theory and modeling of open quantum systems, we will address long-standing questions behind spin decoherence in various condensed matter systems as well as new emerging questions such as the evolution and destruction of entangled states. We will address further technologically relevant questions such as the effect of interfaces on spin coherence in semiconductor devices, as well as the effect of removal or addition of an electron spin (by optical, or electrical means) on the coherent state of coupled nuclear spins. Six graduate students and postdocs will participate in a stimulating international collaboration between Oxford, Heriot-Watt and Princeton, supported by the fluid exchange of young researchers between the participating institutions, as well as interactions with their collaborators around the world. The project partners will continue to host undergraduate students in their laboratory on summer projects connecting with this research proposal. The Oxford and Heriot-Watt teams will continue to participate in enhancing the public understanding of science by, for example, presenting work at the Royal Society Summer Exhibit and hosting local high-school students in their laboratories. The Oxford investigators have experience producing a series of award-winning video podcasts and the collaboration will build on this experience to produce a joint series of podcasts, aimed a general audience, describing the basic science behind this proposal and the exciting applications which it promises.The grant will support and strengthen an existing and highly successful collaboration between researchers at Princeton, Oxford and Heriot-Watt. The groups have a strong track record performing the experiments and developing the techniques which motivate and enable the research proposed here. Together, the investigators bring together a range of expertise, including magnetic resonance (ESR, ENDOR), quantum information theory, density functional theory, semiconductor physics and organic chemistry. The collaborative nature of this proposed activity allows the focus to be on fundamental physical questions spanning a diverse range of physical quantum spin systems, increasing the impact of the experiments and range of beneficiaries in the scientific community. Finally, the complementary instrumentation across the three institutions provides the necessary set of experimental tools required for this challenging experimental program.

There are two natural ways to describe the local structure of a graph or network: by asking what graphs occur as minors, and by asking what graphs occur as induced subgraphs. The theory of graph minors was developed by Robertson and Seymour in an influential series of papers, and gives a very satisfactory picture. However, the picture for induced subgraphs is more complex and much less is known. A central problem in the field is the Erdos-Hajnal conjecture. It has been known since the results of Ramsey and of Erdos and of Szekeres in the 1930s that every graph has a clique or stable set of size at least logarithmic in the number of vertices. However, Erdos and Hajnal conjectured in the 1980s that forbidding any induced subgraph H causes a dramatic jump, resulting in cliques or stable sets of polynomial size. Recently the PI's (joint with two others) settled the smallest open case, which had been a famous question since the problem was first proposed in the 1980's; and even more recently they have made substantial progress on the next smallest case. These two steps both used new techniques, and it is hoped that these techniques will lead to further progress on this and related problems. More broadly, what is the structure that results when some graph H is excluded as an induced subgraph? We don't expect to get a structure that is necessary and sufficient for excluding a graph H, when H is large (this already seems hopeless for graphs H with six vertices); but it is more likely that there is a structure that is necessary for excluding H and sufficient for excluding some larger graph. This would be the first step towards a general structural theory for induced subgraphs. The aim of this proposal is to spend a period of focused collaboration, to work on the Erdos-Hajnal conjecture and related problems, and to build new tools for understanding the structure of graphs with forbidden induced subgraphs. The proposal builds on a substantial existing collaborationbetween the PIs, who have written more than 45 papers together in the last few years. The grant will give them time and resources for a further period of intense collaboration, to drive further progress in this area.