The last fifty years have witnessed tremendous advances in science and technology with a huge impact on society and economy leading to a new information revolution in analogy with the industrial one. Although electronic devices have reached an incredible level of complexity, control and miniaturisation, information processing relies on the same classical principles enunciated by mathematicians in the 1930s (Turing, Church, von Neumann). In the 1980s, visionary ideas from theoretical physicists, including R. P. Feynman and D. Deutsch, and later from computer scientists such as P. Shor, combining concepts from quantum mechanics led to another revolution of information technology: the birth of quantum information theory. In the classical world, a bit, the smallest unit of information, can assume values 0 or 1 corresponding roughly to an electrical circuit being open or closed. In the quantum world, instead, one deals with quantum bits or qubits, embodied for example by an electron spin or a photon polarisation. These qubits can assume the two values 0 and 1 as in the classical case but they can also be prepared in a superposition of the two values simultaneously. This, apparently shocking, property has been verified in numerous experiments and is responsible for the amazing speed-up of certain tasks like integer numbers factorisation with quantum computers, i.e. devices that process qubits in analogy with traditional computers. So far quantum computers have only been realised with a small number of qubits-no more than ten-with trapped ions or neutral atoms, photons but also solid state devices. Large scale quantum computers are therefore expected to be realised only in a few decades. However special purposes quantum computers, called quantum simulators are currently being produced in laboratories working with atoms at temperatures one billionth above the absolute zero (ultracold). Such experiments aim at reproducing, with a controlled environment, the physics of hard to access quantum materials, for example a high-temperature superconductor, thus allowing scientists to probe its properties and test models and theories. A big open question for quantum simulators with ultracold atoms is how, once the sample is prepared in a quantum state, to detect its features. Several techniques are being used based on imaging through a high resolution optical microscope or on scattering of laser light off the sample. In this project we propose the use of a beam of polarised light to probe arrays of neutral atoms. As a consequence of the light-atoms interaction, the light polarisation rotates depending on the state of the atoms. Therefore the outgoing pulse of light, that can be measured, gives information about the state of the atoms. The advantage of this scheme is that one can perform the measurement without destroying the atomic samples as in other proposals. The outcomes of this project will shed light on the intimate structure of the quantum state of many qubits embodied by atoms trapped by electromagnetic fields. For this reason, it is expected to have a strong impact not only in quantum information theory, but also in atomic physics, in statistical mechanics and in the condensed matter physics. Qubits have another peculiarity compared to their classical counterpart: one can correlate the state of one qubit with that of another one in such a way that if one performs a measurement of the two qubits the outcomes always coincide. This phenomenon called entanglement is at the basis of quantum information applications like quantum teleportation. Another goal of this project is a proposal to entangle two of these ultracold atomic samples thus creating entanglement between two separated massive objects composed of hundreds of atoms. The scheme we propose can be implemented in the next generation of experiments with ultracold atoms.

The proposed project focuses on metrics and completions of triangulated categories. The two main objectives are to exploit recent breakthroughs in the theory of metrics on triangulated categories to answer open questions in the representation theory of algebras, and to push their development to the next level. Distance is a fundamental notion which allows us to interpret the world around us. The idea of distance applies across myriad contexts, from distance between physical objects and navigating the space we live in to more conceptual notions of distance in sets of data that provides us with enormous predictive power. Abstracting these disparate incarnations leads to the mathematical notion of a metric space. In his transformative 1973 paper, Lawvere introduced the notion of a metric on a category, by assigning to each morphism a length, and with it a way of measuring how far objects are away from each other, thus linking these fundamental concepts to the categorical world. This provides a potent formalism for simultaneously treating both the distance between objects and how they interact with one another. The proposed project tackles pressing questions relating to the theory of metrics, specifically in triangulated categories. Triangulated categories were introduced more than half a century ago by Verdier in his thesis. With roots and a continuing key role in the fields of algebraic geometry (derived categories of coherent sheaves, motives) and algebraic topology (stable homotopy theory), triangulated categories are crucial to modern day research in a plethora of contexts beyond these subjects, such as in representation theory (derived and stable module categories), symplectic geometry (Fukaya categories), algebraic analysis (Fourier-Sato transform and microlocalisation), and mathematical physics (D-branes and homological mirror symmetry). Given their ubiquity throughout mathematics, it might initially come as a surprise that interesting methods for constructing a new triangulated category from a given one are notoriously elusive. Most recently, Neeman has succeeded in using the technology of metrics and completions to provide a way to obtain a new triangulated category from a triangulated category with a "good" metric. Considering the scarcity, and relative restrictiveness, of previously known methods for constructing a new triangulated category from a given one, the potential of this result is immense. In particular, being able to produce new triangulated categories has the potential to impact several conjectures, particularly in noncommutative motives. The goal of the proposed project is twofold: To further the theory of metrics and completions of triangulated categories and to exploit it to advance our understanding of the representation theory of finite dimensional algebras. In light of the new and interesting way of constructing triangulated categories via metrics and completions, and the dream of an explicit computation of these at our fingertips, we use combinatorial models on the one hand, and dg enhancements on the other to provide machinery to make this become a reality. At the same time, we exploit the theory of metrics and completions to allow for a fresh approach to study the poset of t-structures, with an emphasis on determining precisely under what circumstances this poset forms lattice.

This proposal seeks to develop a fundamentally new multiscale framework for data-adaptive exploratory analysis of multivariate real-world processes. This will be achieved through a rigorous treatment of both within- and cross-channel intrinsic signal features, spanning time, space, frequency and entropy. Particular emphasis will be on approaches that are free of statistical assumptions and mathematical artefacts, and match the time-varying oscillatory modes inherent in multivariate data. This will help bypass the mathematical obstacles associated with currently used techniques (Fourier, wavelet), which rely on fixed basis functions and integral transforms, thus colouring the representation, limiting their accuracy, and restricting their applicability in problems involving real-world drifting and noisy information. For multiscale data current statistical and information theoretic measures are inadequate, as they will indicate high correlation for two data channels that share common noises, but do not contain the same useful signal. The proposed data-adaptive analysis framework will resolve such issues, and will create natural "intrinsic" data association measures (intrinsic multi-correlation, intrinsic multi-information). While current univariate data-adaptive approaches have enormous potential, they are not suitable for direct application to multivariate or heterogeneous sources, as they are bound to create a different number of basis functions for every data channel. Wearable systems, such as bodysensor networks, strive to find a balance between performance and user benefits (low cost, ease of use), and require next-generation signal processing tools to establish the extent to which they can produce valuable information. The thrust of this proposal is on developing rigorous, data-adaptive, compact, and physically meaningful signal processing solutions in order to provide an algorithmic support for progress in multi-sensor and wearable technologies. Our own initial multivariate data-adaptive solutions show great promise; they need to be further developed and comprehensively tested for data exhibiting rotation-dependent (noncircular) distributions, power imbalance, uncertainty, and noise. With the aid of nonlinear optimisation in the algorithmic design and insights from dynamical complexity science and multiresolution information theory, our approach promises a quantum step forward in multivariate data analysis, and a significant long-term impact. The successful outcomes of this proposal will open radically new possibilities for advances in areas that depend on multi-sensor data, and a new front of research in applications dealing with uncertainty, noncircularity, complexity, and nonstationarity in multi-channel recordings. To maximise the short- to medium-term impact of this work and for cost effectiveness, we consider applications in emerging wearable technologies for brain monitoring, in collaboration with the Royal Brompton Sleep Clinic in London and Aarhus University in Denmark.

Summary The Rio Summit of 1992 propelled biodiversity into a global spotlight pointing to tremendous human-induced species losses in the Earth's ecosystems. Now there is an urgent need to advance our knowledge on how and why species disappear from ecosystems and the implications of these losses for important goods and services that we rely on (e.g. drinking water, food, spiritual values). One crucial landscape feature thought to have a major influence on biodiversity is connectivity - how connected habitats within the landscape are with one another. A key issue here is alteration of our natural landscapes via the creation of roads, towns and farmland. Under such circumstances natural habitats become isolated and degraded which impedes the dispersal of native species. We believe, based on preliminary evidence, that ease of dispersal across the landscape is a key feature that reduces rates of species loss in human-affected ecosystems thus preserving high biodiversity and valuable (monetary and cultural values) ecosystem services. Lakes are uniquely useful for examining questions about biodiversity, connectivity and ecosystem services as they permit long-term (over centuries) changes in biodiversity to be studied through the analysis of fossil remains in sediment cores. The majority of aquatic organisms (e.g. algae, plants, invertebrates) leave identifiable parts in sediments, which can be dated to reveal a history of ecological change. In the proposed study we will focus on two UK lake districts: the Norfolk Broads, England and the Upper Lough Erne (ULE) lakes, Northern Ireland. Both contain numerous (60+) shallow lakes, have a long history of agricultural pollution (50-100 years) and have been subject to invasions of non-native species (notably zebra mussels). However, the Broads are mostly highly degraded, having generally turbid water with few plants, while the ULE lakes have generally clear waters and abundant and species-rich plant beds. We propose that this key difference relates to elevated connectivity amongst the ULE lakes due to a higher density of linking channels and the occurrence of winter floods which cover much of the system. This, we believe, enhances the exchange of plants and plant seeds, which in turn buffers against permanent plant extinctions in individual lakes, despite pressures from pollution. Through the collation and collection of data on present-day water plant abundance and diversity in many individual lakes in these two systems, analysis of the amounts of carbon, nitrogen and phosphorus taken up by plants in these lakes, and by analysing sediment cores to detect changes in aquatic plant diversity and pollution over time, our research will address the following key questions: 1. Does higher connectivity buffer biodiversity loss in the face of pollution and species invasions? 2. How do changes in water plant diversity affect key functions of lake ecosystems that in turn influence the services they provide to humans? 3. Can knowledge gained from questions 1 and 2 be translated into changed conservation practices to reduce biodiversity and ecosystem service losses from aquatic landscapes? Our project will give policy makers and conservation organisations vital information to inform landscape planning, such as the need to prioritise protection of existing high biodiversity areas (e.g. species-rich lakes) and to maintain connectivity of such sites with others. We anticipate generating an evidence-base that will argue for the maintenance or enhancement of connectivity to increase the resilience of our ecosystems to future biodiversity loss. In a world threatened by climate change, habitat fragmentation and pollution, knowledge of the relationships between dispersal, biodiversity and key ecosystem services is essential to our well being.

Many biological processes are based on chemical reactions. Viscosity determines how fast molecules can diffuse, and react. Therefore in cells viscosity can affect signalling, transport and drug delivery, and abnormal viscosity has been linked to disease and malfunction. In spite of its importance, measuring viscosity on a scale of a single cell is a challenge. Traditionally used mechanical methods are no longer applicable and must be substituted by a spectroscopic approach. Such spectroscopic approaches exist, e.g. single particle tracking, monitoring the rate of fluorescence recovery after photobleaching, or monitoring the rate of viscosity-dependent photochemical reactions. However all of the above are single point measurements and in a complex heterogeneous environment of a cell can not provide full information. The spectroscopic approach which allows imaging or mapping of viscosity would be of great benefit. This proposal aims to measure and map viscosity inside a single cell with high precision and high spatial resolution using novel fluorescent probes, called molecular rotors. In molecular rotors fluorescence competes with intramolecular rotation. In a viscous environment rotation is slowed down and this strongly affects fluorescence. Thus viscosity can be measured by detecting the change in either the fluorescence spectra or lifetimes. Existing technology allows imaging of either the fluorescent spectra or lifetimes with excellent spatial resolution in single live cells. To date we have produced maps of viscosity in certain parts of cells using this approach and demonstrated that local viscosity in those compartments can be up to 100x higher than that of water.Important advantage of molecular rotor approach is a very short measurement time. Using this advantage, this proposal aims to monitor how viscosity in a cell changes during dynamic biological processes, e.g. change in the membrane structure upon cell perturbation, drug administration and cell death.Photodynamic therapy (PDT) is a form of cancer treatment, which relies on the generation of short-lived toxic agents within a cell upon irradiation of a drug. The efficacy of this treatment critically depends on the viscosity of the medium through which the cytotoxic agent must diffuse during its short life span. This proposal will monitor how cell viscosity and other vital biophysical cell parameters change during PDT. The novelty of our approach is in using spatially resolved irradiation of the drug within cells. E.g. we can irradiate a single organelle and monitor the change in the entire cell. Alternatively, we can irradiate the group of cells and monitor the behaviour of its neighbours. This approach is ideal tool to directly probe the 'bystander effect', when the cells which have not been directly treated show significant response to therapy, the effect which is very important in radiation and PDT cancer treatment. This proposal will be carried out in the Chemistry Department at Imperial College London where multidisciplinary collaborations are established to ensure the success of the work proposed. This project will address both the fundamental scientific issues in photochemistry and cell biology and also encourage the development of applications, such as measuring viscosity as a diagnostic tool and for monitoring the progress of treatments.