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.

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.

Complex interactions between turbulent convection and stably-stratified flows in the presence of background rotation are important for a wide range of problems in various engineering contexts, in atmospheric and oceanic science, and in stellar and planetary astrophysics. This project will investigate the nature of flows between a heat source and a heat sink that are displaced both vertically and horizontally relative to each other, in the presence of strong background rotation. In the absence of background rotation, if a heat source is located at a lower altitude than the sink, one would generally expect a strongly convective circulation to result, carrying heat directly and vigorously from the source to the sink. With background rotation, however, evidence from experiments, simulations and in geophysical flows suggest that the resulting circulation may spontaneously partition itself into a convectively unstable/neutral region (where temperature becomes well mixed and doesn't vary much with height) that interacts with a statically stable, so-called baroclinic region (where temperature increases with height and develops a thermal gradient from one side to the other). Wave-like instabilities may develop within this baroclinic zone that may play a crucial role in stabilising the vertical stratification and dominating the transfer of heat and momentum where they occur. Moreover, there is evidence to suggest that if the transport of heat by the instability acts more rapidly than other heat exchange processes, this stabilizing effect may act within a nonlinear feedback loop, somewhat like a thermostat, adjusting the flow back towards a weakly nonlinear/unstable 'critical' state - sometimes referred to as 'self-organized criticality'. Such strongly nonlinear and convective motions are difficult to model accurately, however, so the mechanisms involved, though probably ubiquitous in certain engineering systems and in nature, are not well understood. We therefore propose to set up an experimental configuration which entails heating a body of fluid in a cylindrical container on a rotating platform along an annular ring at the bottom of the tank close to the outer radius, and cooling it through a circular disk near the centre of the tank at the upper surface. Preliminary numerical simulations and experiments (carried out in my group and with proposed collaborators in the USA and Spain) already suggest that such flows will readily form a statically stable (though baroclinically unstable) zone between convectively unstable regions over/underlying the heated or cooled boundaries. We therefore plan to measure the characteristics of the resulting flows through combinations of in situ thermal sensors and particle image velocimetry (PIV) techniques, including the innovative possibility of using thermochromic liquid crystal particles to determine velocities and temperatures simultaneously within the flow. This will facilitate the determination of flow structures, heat and momentum transports within the flow, and to characterize the development of any kinetic energy cascades that may emerge as more turbulent regimes are explored. The idealised nature of these experiments should ensure that the results obtained will be applicable to a wide variety of problems in various disciplines. Such a configuration may be seen as an idealisation of a variety of industrial processes (e.g. in rotating semiconductor crystal growth melts, process mixing techniques in chemical engineering, convective flows in turbomachinery etc.), and of a number of geophysical and astrophysical problems in which stably and unstably stratified flows interact in the presence of background rotation. These include the Earth's atmosphere and climate system and its response to variations in its radiative heating and cooling, other planetary atmospheres (notably Mars, Venus and the gas giant planets), and in stellar interiors (e.g. the tachocline region within the Sun).