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In homotopy theory, topological spaces (i.e. shapes) are regarded as being the same if we can deform continuously from one to the other. Algebraic varieties are spaces defined by polynomial equations, often over the complex numbers; studying their homotopy theory means trying to tell which topological spaces can be deformed continuously to get algebraic varieties, or when a continuous map between algebraic varieties can be continuously deformed to a map defined by polynomials.If the polynomials defining a variety are rational numbers (i.e. fractions), this automatically gives the complex variety a group of symmetries, called the Galois group. Although these symmetries are not continuous (i.e. nearby points can be sent far apart), they preserve something called the etale topology. This is an abstract concept which looks somewhat unnatural, butbehaves well enough to preserve many of the topological features of the variety. Part of my project will involve investigating how the Galois group interacts with the etale topology. I also study algebraic varieties in finite and mixed characteristics. Finite characteristics are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. For instance, in characteristic 3 the equation 1+1+1=0 holds. In mixed characteristic, p need not be 0, but the sequence 1,p, p^2, p^3 ... converges to 0.Although classical geometry of varieties does not make sense in finite and mixed characteristics, the etale topology provides a suitable alternative, allowing us to gain much valuable insight into the behaviour of the Galois group. This is an area which I find fascinating, as much topological intuition still works in contexts far removed from real and complex geometry. Indeed, many results in complex geometry have been motivated by phenomena observed in finite characteristic.Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to those of algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry. To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can themselves have symmetries and so on, giving rise to infinity stacks.Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities (points where the space is not smooth), and regions of different dimensions. Partially inspired by ideas from theoretical physics, it has been conjectured that every moduli stack can be extended to a derived moduli stack, which would have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness . Different classification problems can give rise to the same moduli stack, but different derived moduli stacks. Much of my work will be to try to construct derived moduli stacks for a large class of problems. This has important applications in algebraic geometry, as there are many problems for which the moduli stacks are unmanageable, but which should become accessible using derived moduli stacks. I will also seek to investigate the geometry and behaviour of derived stacks themselves.A common thread through the various aspects of my project will be to find ways of applying powerful ideas and techniques from a branch of topology, namely homotopy theory, in contexts where they would not, at first sight, appear to be relevant.

Energy efficient processes are increasingly key priorities for ICT companies with attention being paid to both ecological and economic drivers. Although in some cases the use of ICT can be beneficial to the environment (for example by reducing journeys and introducing more efficient business processes), countries are becoming increasingly aware of the very large growth in energy consumption of telecommunications companies. For instance in 2007 BT consumed 0.7% of the UK's total electricity usage. In particular, the predicted future growth in the number of connected devices, and the internet bandwidth of an order of magnitude or two is not practical if it leads to a corresponding growth in energy consumption. Regulations may therefore come soon, particularly if Governments mandate moves towards carbon neutrality. Therefore the applicants believe that this proposal is of great importance in seeking to establish the current limits on ICT performance due to known environmental concerns and then develop new ICT techniques to provide enhanced performance. In particular they believe that substantial advances can be achieved through the innovative use of renewable sources and the development of new architectures, protocols, and algorithms operating on hardware which will itself allows significant reductions in energy consumption. This will represent a significant departure from accepted practices where ICT services are provided to meet the growing demand, without any regard for the energy consequences of relative location of supply and demand. In this project therefore, we propose innovatively to consider optimised dynamic placement of ICT services, taking account of varying energy costs at producer and consumer. Energy consumption in networks today is typically highly confined in switching and routing centres. Therefore in the project we will consider block transmission of data between centres chosen for optimum renewable energy supply as power transmission losses will often make the shipping of power to cities (data centres/switching nodes in cities) unattractive. Variable renewable sources such as solar and wind pose fresh challenges in ICT installations and network design, and hence this project will also look at innovative methods of flexible power consumption of block data routers to address this effect. We tackle the challenge along three axes: (i) We seek to design a new generation of ICT infrastructure architectures by addressing the optimisation problem of placing compute and communication resources between the producer and consumer, with the (time-varying) constraint of minimising energy costs. Here the architectures will leverage the new hardware becoming available to allow low energy operation. (ii) We seek to design new protocols and algorithms to enable communications systems to adapt their speed and power consumption according to both the user demand and energy availability. (iii) We build on recent advances in hardware which allow the block routing of data at greatly reduced energy levels over electronic techniques and determine hardware configurations (using on chip monitoring for the first time) to support these dynamic energy and communications needs. Here new network components will be developed, leveraging for example recent significant advances made on developing lower power routing hardware with routing power levels of approximately 1 mW/Gb/s for ns block switching times. In order to ensure success, different companies will engage their expertise: BT, Ericsson, Telecom New Zealand, Cisco and BBC will play a key role in supporting the development of the network architectures, provide experimental support and traffic traces, and aid standards development. Solarflare, Broadcom, Cisco and the BBC will support our protocol and intelligent traffic solutions. Avago, Broadcom and Oclaro will play a key role in the hardware development.

We propose to create in the UK a novel research capability providing Angstrom Analysis for dynamic in-situ reaction studies under controlled conditions of temperature and continuous gas atmosphere rather than the usual high vacuum. The new design provides the world's first full function aberration corrected environmental scanning transmission electron microscope (AC ESTEM). In association with partners in the vibrant UK chemical and energy industries we will generate fundamental application science underpinning nanoparticle based solid state heterogeneous catalysis used in gas-solid reactions. We will modify an existing AC TEM/STEM instrument to complement and extend with gas reaction studies the National AC STEM Facility's superior image and energy resolutions in high vacuum. It will be used in York programmes and collaborative projects with other groups through the AC STEM. It builds on the PIs' established reputations for global leadership in ETEM, with most of the worldwide activity to date - all overseas - based on >10 high resolution ETEMs and many of them AC (on the TEM image side only), using core technology from the authors' earlier developments. Preliminary 'proof-of-principle' has been demonstrated on the remotely controlled double aberration corrected JEOL 2200FS TEM/STEM at York; combining sub-Angstrom (<0.1nm) resolution, unrestricted HAADF Z-contrast STEM imaging, wide angle electron diffraction and EDX (+ EELS) chemical analysis not available on ETEMs. The double aberration correction collects, in a single and often directly interpretable TEM image, a full range of spatial frequencies at close to zero defocus to minimise image delocalisation at internal interfaces such as grain boundaries, external surfaces, defects and other key discontinuities. This is especially important for dynamic in-situ studies with continuously changing data making impractical older through-focal series reconstruction methods. AC also transforms the sensitivity of STEM analysis. The work will use analytical methods established with 'frozen' and process extracted samples, and apply them to the study of continuous processes at new levels of sensitivity and relevance. Access to key intermediate states and phases may be critical to understand and control process mechanisms; but they may be metastable with respect to conditions, including temperature or chemical environment, and therefore not accessible through ex-situ or pulse studies. A very practical example, in which there is leading UK industry interest and support, is the nano-structure and related property stability of supported metal nanoparticle heterogeneous catalysts. Through synthesis, activation, operation, deactivation, reactivation and recovery mechanisms, understanding at a fundamental level is critical for managing on a rational basis industrial practice for sustained activity and selectivity; and where necessary recovering these key attributes when lost. The project direction is closely aligned with the domain science needs of real world academic and industrial applications, and there are early adoption prospects for underpinning key technologies; including to extend useful process life cycles. For example, this is critical for the wider commercial viability of fuel cells. The proposal has the support of leading UK companies in the vibrant and internationally competitive chemical industry sector, and of academic collaborators. At the same time, the new learnings in basic domain science are also directed towards opening up new applications of pressing societal value in the environment. Fundamental physical science research with strategic and tactical industrial applications leads to differentiated intellectual products with an initiative unique in the UK and fully competitive globally. The project will extend and apply core nanoparticle catalysis science and technology, and train a new cohort of students, postdocs, senior staff and visitors.

Free boundary problems are challenging mathematical problems which involve solving nonlinear problems in domains whose shapes have to be found as part of the solution. They occur in many industrial, environmental and biological applications. Examples are coating problems, ocean waves and tumors. In this proposal we concentrate on hydroelastic waves and in particular on applications to ice sheets. The research is motivated by the understanding of man-made large floating structures (especially airports) and by Antarctic exploration where often heavy equipment will travel over roads on floating ice and aircraft will operate on floating ice-sheet runways. Waves under ice sheets have also recently attracted attention because they are one of the many factors that need to be considered as having an impact in climate change. The mathematical formulation of hydroelastic waves involve complicated systems of nonlinear integro-differential equations for which most studies have been restricted to linear models. Although linear approximations are often adequate, there are many situations in which nonlinearities and large defections of the ice sheet cannot be neglected. Therefore we propose to develop fully nonlinear numerical theories and weakly nonlinear asymptotic models to tackle these problems. Solitary waves, dark solitons, internal waves and three dimensional waves are among the topics to be studied. The previous experience of the PI, CIs and visiting researcher with nonlinear waves will be very instrumental for the success of the project. The intended RA is Dr Leonardo Xavier Epsin. His strong background in modelling, asymptotics and numerical simulations makes him very well suited to work on the proposed problems. In case he were not able to take the position, other qualified candidates are known to the Pi and CIs.

WAM is a project which started from asking a question about what would make a transformational difference to someone who is experiencing difficulties in walking. The answer was to be able to walk without visible (or audible) assistance. Following some initial work in an EPSRC project called RELEASE, WAM is starting on the road to developing just this sort of assistive technology: an exoskeleton that can be worn and that can also act as a muscle so that it can support the whole walking cycle, providing support and control to the user and enabling them to walk otherwise unaided. We have selected some of the likely possibilities from those explored during the RELEASE project and WAM will develop these further, combine them and test them against the strength, stiffness, flexion and strain requirements of the walking process. Three technologies will be explored in parallel: (1) using Vanadium Oxide (V2O5) as a chemical actuator, (2) using magnetic gels as a dynamic controllable mechanism and (3) using interlockable ceramic tiles as a surface medium. V2O5 is able to flex when exposed to an electrostatic potential and can exhibit strength (about 10 times the strength found in skeletal muscle) so this would seem to be a useful substance to use as the basis for an actuator. Macroporous magnetic gels can be used to compress rapidly under a magnetic field to deliver drugs (by squeezing the drugs from the pores as the material compresses under the influence of an induced magnetic field). Interlockable ceramic tiles can provide the stiffness needed for the rigidity needed by a skeleton, but once unlocked an allow the structure to bend in a required direction. These will be tested separately and in combination to see if the V2O5 working with the magnetic gel could act as a sufficiently strong actuator and key for the locking/unlocking mechanism so that the material can demonstrate sufficient strength, stiffness and strain capabilities to be worth scaling up in a future project to assist walking. Other possibilities would also be available from such material - it does not have to be used only for walking as it could be used for other joints which can need support but which also need to bend - elbow, wrist, ankle as well as knee are all candidates for such support. Also, it might provide an interesting support where it is desirable to make the assistance variable - when the person needs support and when they would benefit from being encouraged to take on the activity themselves. A variably flexible material of this sort would therefore be of use in conditions such as limb fractures (where absolutely fixed support is essential at one point on the process, but rehabilitation of associated muscles would be beneficial as the facture is healing, but is still weak, before the fixed support can be removed) or conditions such as Carpal Tunnel Syndrome where support needs might vary. While the technological work is underway we will be working with a patients' group and a group of clinicians to understand more about what end users might be looking for in their assistive technology and how this particular type of support might be useful for them. It is important to realise that this is a novel way of providing dynamic support for activities such as walking and thus there is a sense of all sides needing to understand how best to use the technology as it emerges from the laboratory. The present project will not deliver the full working prototype of a walking support system, but it will develop, test and show what can be done in terms of the material and its control system and the extent to which this approach to actuation of locomotory support could be achieved. The final tests will show how much stiffness and strength it can deliver and thus whether or not this is the right way to proceed in further projects.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Computer simulations of fluid flow are playing an increasingly important role in aerodynamic design of numerous complex systems, including aircraft, cars, ships and wind turbines. It is becoming apparent, however, that for a wide range of flow problems current generation software packages used for aerodynamic design are not fit for purpose. Specifically, for scenarios where flow is unsteady (highly separated flows, vortex dominated flows, acoustics problems etc.) current generation software packages lack the required accuracy; since they are ubiquitously based on 'low-order' (first- or second-order) accurate numerical methods. To solve challenging unsteady flow problems, and remove the need for expensive physical prototyping, newer software based on advanced 'high-order' accurate numerical methods is required. Additionally, this software must be able to achieve high-order accuracy on so-called 'unstructured grids' - used to mesh complex engineering geometries, and it must be able to make effective use of next-generation 'many-core' computing hardware (such as Nvidia Tesla GPUs, Intel Xeon Phi Co-Processors, and AMD FirePro GPUs), which will likely underpin future HPC platforms. Advanced high-order Flux Reconstruction (FR) methods, combined with many-core accelerators, could provide a `gamechanging' technology capable of performing currently intractable unsteady turbulent flow simulations within the vicinity of complex engineering geometries. However, various technical issues still need to be addressed before the above technology can be used `in anger' to solve real-world flow problems, which often involve `sliding planes' (situations when two computational meshes slide across one another in a non-conforming fashion). The key objectives (of the academic component) of the proposal are to develop a treatment for sliding planes that works effectively with FR methods on manycore accelerators, and demonstrate the performance of FR methods on many-core accelerators for a range of industry led test cases proposed by the financial (CFMS and Zenotech) and non-financial (Airbus, EADS, BAE, Rolls-Royce, ARA, UK Aerodynamics Centre) project partners. The academic component of the proposal will be lead by Dr. Peter Vincent (a Lecturer in the department of Aeronautics at Imperial College London), and will build upon current work funded by 3 x EPSRC DTAs, 1 x Airbus/EPSRC iCASE DTA, and an EPSRC Early Career Fellowship (EP/K027379/1).

The Warwick EPSRC mathematics symposium is organised annually by the University of Warwick with the support of the EPSRC for the benefit of the mathematical sciences community in the UK. It brings leading national and international experts together with UK researchers in a year-long programme of research activities focused on an emerging theme in the mathematical sciences. The proposed symposium for the 2015-16 academic year will concentrate on the theme of "Fluctuation-driven phenomena and large deviations". In very general terms, the symposium will constitute an interdisciplinary focus on understanding the consequences of the interplay between stochasticity and nonlinearity, a recurrent challenge in many areas of the mathematical sciences, engineering and industry. Stochastic processes play a fundamental role in the mathematical sciences, both as tools for constructing models and as abstract mathematical structures in their own right. When nonlinear interactions between stochastic processes are introduced, however, the rigorous understanding of the resulting equations in terms of stochastic analysis becomes very challenging. Mean field theories are useful heuristics which are commonly employed outside of mathematics for dealing with this problem. Mean field theories in one way or another usually involve replacing random variables by their mean and assuming that fluctuations about the mean are approximately Gaussian distributed. In some cases, such models provide a good description of the original system and can be rigorously justified. In many cases they do not. Understanding the latter case, where mean-field models fail, is the central challenge of this symposium. We use "fluctuation driven phenomena" as a generic term to describe the kinds of effects which are observed when mean field theories fail. The challenges stem from the fact that the rich phenomenology of deterministic nonlinear dynamics (singularities, nonlinear resonance, chaos and so forth) is reflected in the stochastic context by a variety of interesting and sometimes unintuitive behaviours: long range correlations, strongly non-Gaussian statistics, coherent structures, absorbing state phase transitions, heavy-tailed probability distributions and enhanced probabilities of large deviations. Such phenomena are found throughout mathematics, both pure and applied, the physical, biological and engineering sciences as well as presenting particular problems to industrialists and policymakers. Contemporary problems such as the forecasting of extreme weather events, the design of marine infrastructure to withstand so-called "rogue waves", quantifying the probability of fluctuation driven transitions or "tipping points" in the climate system or estimating the redundancy required to ensure that infrastructure systems are resilient to shocks all require a step change in our ability to model and predict such fluctuation-driven phenomena. The programme of research activities constituting this symposium will therefore range from the very theoretical to the very applied. At the theoretical end we have random matrix theory which has recently emerged as a powerful tool for analysing the statistics of stochastic processes which are strongly non-Gaussian without the need to go via perturbative techniques developed in the physical sciences such as the renormalisation group. At the applied end we have questions of existential importance to the insurance industry such as how to cost the risk of extreme natural disasters and quantify their interaction with risks inherent in human-built systems. In between we have research on the connections between large deviation theory and nonequilibrium statistical mechanics, extreme events in the Earth sciences, randomness in the biological sciences and the latest numerical algorithms for computing rare events, a topic which has seen strong growth recent years.

Methods for the formation of C-C and C-Y (Y = O, N) bonds are crucial for the synthesis of new molecules. In the last 20 years there have been huge advances in Pd-catalysed cross-coupling reactions and this was recognized in the award of the 2011 Nobel Prize. These reactions involve joining together two organic molecules, one of which features a bond between carbon and a halogen (a C-X bond) while the other coupling partner may feature a non-carbon centre such as tin, boron or zinc (a C-M bond). The coupling of these two partners to give a new C-C bond usually involves a palladium catalyst. Later versions involved forming C-N or C-O bonds. This coupling reaction generally works well, however the efficiency of this individual step masks significant waste of time and energy as well problems with environmental sustainability. These problems arise because both coupling partners ultimately derive from precursors that only contain C-H bonds and therefore require prior synthesis that usually involves several steps each with costly energy and purification implications. Moreover the final coupling process itself eliminates salts that must first of all be separated from the reaction products (a further costly and expensive process) before disposal, often with significant environmental impact. A far more desirable approach would be to use the unactivated C-H containing precursors directly as the coupling partners. Such compounds are readily available and cheap. This approach would circumvent the need for the costly and wasteful preactivation that is required to make C-X and M-C species, as well as the post processing clean up of M-X by-products. Until very recently this approach has not been adopted as C-H based precursors are usually rather chemically inert. However, catalysis based on such C-H species (catalytic C-H activation) is now within reach, mainly due to recent advances where the means to activate the C-H bond with a transition metal catalyst have been understood. A key point in C-H activation is to have a directing group elsewhere on the feedstock molecule so that it can interact with the metal catalyst and so bring the C-H bond close enough to react. The M-C bond that is thus formed can then undergo reactions with other substrates to produce the desired C-C, C-N or C-O bond. Moreover, if instead this new bond is formed with another atom in the same molecule then a ring is formed. Such cyclic compounds containing an N or O atom are heterocyclic compounds and these play a key role as major constituents of pharmaceuticals and agrochemicals. In addition they often have interesting optical and electrical properties in their own right that are important in a range of technological applications. It is therefore crucial that the synthesis of heterocyclic compounds is as efficient as possible; moreover the need for a wide range of heterocycles with different properties depends upon the development of new, efficient methods for their synthesis. There are several precedents for this type of catalytic C-H activation in the scientific literature, however to date there is little understanding of what controls this reactivity. Thus the range of species that can be made is limited, the catalysis is not yet efficient and the selectivity of the reaction is poorly understood. To improve this situation requires a deeper understanding of how these systems work. We aim to provide this here through a combination of experimental studies and computational modeling. By understanding the factors that control reactivity and selectivity we will be able to design new, more efficient catalysts and also to widen the scope of the catalytic C-H activation methodology. The ultimate aim is to provide a flexible set of efficient synthetic tools that chemists will be able to use to make a wide range of important heterocycles in an environmentally sustainable manner.

Nanotechnology is a significant enabling research activity at both a national and international level. Concerned with the manipulation and arrangement of material on the nanometre-scale, the transformative possibilities of this field are immense for the physical, biological and medical sciences and their allied industrial and clinical applications. The universities of Leeds, York and Sheffield have an exceptionally strong international record of research and research-led teaching in nanotechnology and represent a strong regional focus in the UK in this field. Much research is carried out collaboratively and inter-disciplinarily in well-resourced and sustainably managed facilities.However, despite the strength of such existing activities, it is clear that certain capabilities require an urgent and substantial transformation if we are to continue to offer internationally competitive research in this field over the next decade. Specifically - and the objective of this proposal - there is a pressing need to establish a state-of-the-art electron-beam lithography machine for fabrication of structures with a <10 nm resolution, with highly reproducible stitching and overlay accuracy <20 nm. The proposed facility would not only be unique in the region, but will also be leading both in the UK and internationally. It will meet the future needs of researchers over the next decade and beyond, allow us to capitalise on previous investments, grow research income from a wide variety of sources, attract and retain the highest calibre staff in the UK, and build a capability to develop a skill-set for ambitious, adventurous and transformative research, and exploitation. Furthermore, it will act as a focus in the region, drawing in researchers from industry and other universities for collaborative programmes. Such direct engagement with industry will open up routes for further investment as well as exploitation of new science and technology. A wide range of research will benefit, much cross-disciplinary; immediate exemplars, drawing upon proven track records of the investigators, include research into nanomagnetism, spintronics, bio-nanotechnology, nanoelectronics, single-molecule devices, and high-frequency electronics, inter alia. During this programme, the facility will be used to support both a range of existing grants, and to underpin future grants, many of which cannot be contemplated without the planned enhancement in capability.Significant contributions to this project (41% of the overall project value) have been secured from the University of Leeds, where the new facility will be based, Yorkshire Forward (the regional development agency), and the electron beam lithography instrument manufacturer. The latter two contributions will be combined to provide funding for 10 PhD studentships to aid uptake of the instrument from researchers across the region, enable pump-priming proving research to be carried out, draw industrial involvement into the project, and increase the availability of skilled personnel at a world leading level to facilitate high technology development. The strong industrial support for this programme is evidenced by letters of intent provided both by international companies (eg Hitachi, Intel, Seagate, Toshiba), and local SMEs (eg Aptuscan).