The main goal of typing is to prevent the occurrence of execution errors during the running of a program. Milner formalised the idea, showing that ``well-typed programs cannot go wrong''. In practice, type structures provide a fundamental technique of reducing programmer errors. At their strongest, they cover most of the properties of interest to the verification community. A major trend in the development of functional languages is improvement in expressiveness of the underlying type system, e.g., in terms of Dependent Types, Type Classes, Generalised Algebraic Types (GADTs), Dependent Type Classes and Canonical Structures. Milner-style decidable type inference does not always suffice for such extensions (e.g. the principal type may no longer exist), and deciding well-typedness sometimes requires computation additional to compile-time type inference. Implementations of new type inference algorithms include a variety of first-order decision procedures, notably Unification and Logic Programming (LP), Constraint LP, LP embedded into interactive tactics (Coq's eauto), and LP supplemented by rewriting. Recently, a strong claim has been made by Gonthier et al that, for richer type systems, LP-style type inference is more efficient and natural than traditional tactic-driven proof development. A second major trend is parallelism: the absence of side-effects makes it easy to evaluate sub-expressions in parallel. Powerful abstraction mechanisms of function composition and higher-order functions play important roles in parallelisation. Three major parallel languages are Eden (explicit parallelism) Parallel ML (implicit parallelism) and Glasgow parallel Haskell (semi-explicit parallelism). Control parallelism in particular distinguishes functional languages. Type inference and parallelism are rarely considered together in the literature. As type inference becomes more sophisticated and takes a bigger role in the overall program development, sequential type inference is bound to become a bottle-neck for language parallelisation. Our new Coalgebraic Logic Programming (CoALP) offers both extra expressiveness (corecursion) and parallelism in one algorithm. We propose to use CoALP in place of LP tools currently used in type inference. With the mentioned major developments in Corecursion, Parallelism, and Typeful (functional) programming it has become vital for these disjoint communities to combine their efforts: enriched type theories rely more and more on the new generation of LP languages; coalgebraic semantics has become influential in language design; and parallel dialects of languages have huge potential in applying common techniques across the FP/LP programming paradigm. This project is unique in bringing together local and international collaborators working in the three communities. The number of supporters the project has speaks better than words about the timeliness of our agenda. The project will impact on two streams of EPSRC's strategic plan: "Programming Languages and Compilers" and "Verification and Correctness". The project is novel in aspects of Theory (coalgebraic study of (co)recursive computations arising in automated proof-search); Practice (implementation of the new language CoALP and its embedding in type-inference tools); and Methodology (Mixed corecursion and parallelism).

Inverse problems are concerned with the reconstruction of the causes of a physical phenomena from given observational data. They have wide applications in many problems in science and engineering such as medical imaging, signal processing, and machine learning. Iterative methods are a particularly powerful paradigm for solving a wide variety of inverse problems. They are often posed by defining an objective function that contains information about data fidelity and assumptions about the sought quantity, which is then minimised through an iterative process. Mathematics has played a critical role in analysing inverse problems and corresponding algorithms. Recent advances in data acquisition and precision have resulted in datasets of increasing size for a vast number of problems, including computed and positron emission tomography. This increase in data size poses significant computational challenges for traditional reconstruction methods, which typically require the use of all the observational data in each iteration. Stochastic iterative methods address this computational bottleneck by using only a small subset of observation in each iteration. The resulting methods are highly scalable, and have been successfully deployed in a wide range of problems. However, the use of stochastic methods has thus far been limited to a restrictive set of geometric assumptions, requiring Hilbert or Euclidean spaces. The proposed fellowship aims to address these issues by developing stochastic gradient methods for solving inverse problems posed in Banach spaces. The use of non-Hilbert spaces is gaining increased attention within inverse problems and machine learning communities. Banach spaces offer much richer geometric structures, and are a natural problem domain for many problems in partial differential equation and medical tomography. Moreover, Banach-space norms are advantageous for preservation of important properties, such as sparsity. This fellowship will introduce modern optimisation methods into classical Banach space theory and its successful completion will create novel research opportunities for inverse problems and machine learning.

The vast, remote seas which surround the continent of Antarctica are collectively known as the Southern Ocean. This region with its severe environment of mountainous seas, winter darkness, strong winds, freezing temperatures and ice is unsurprisingly one of the least explored and under-observed parts of the global ocean. However, because of these extremes, it plays a large and still unquantified role in Earth's climate system. In this region, large amounts of heat and carbon dioxide are exchanged between the atmosphere and the ocean. The physical mechanisms controlling these atmosphere-ocean exchanges are the subject of the NERC ORCHESTRA programme. We propose within PICCOLO to concentrate on the role that chemistry and biology play within those exchanges. In particular, PICCOLO will focus on understanding the mechanisms that transform the carbon contained in the seawater as it rises to the surface near Antarctica, interacts with the atmosphere, ice, phytoplankton and zooplankton inhabiting the near surface, before descending to the ocean depths. PICCOLO will undertake an ocean research expedition to the region close to Antarctica, as computer models and satellite images show that these are areas crucial for carbon processes. Freezing seawater in these regions releases salt into the water below, making it denser and therefore causing it to sink. Strong winds cause the sea ice to be pushed away from the Antarctic coastline, leaving areas of open water called polynyas. Within the polynyas the water has enough light during the summer to allow phytoplankton to grow, as well as providing dense waters which sink to the deep, driving a giant ocean conveyor belt which has a large impact upon Earth's climate system. The PICCOLO team will measure the key variables that control the biological and chemical processes in this region including iron, nutrients, phytoplankton and zooplankton. Crucially the team will study the controlling rate terms between different parts of this biological and chemical system. The PICCOLO team will make use of the latest technologies, including autonomous submarines, gliders and floats, to observe these processes in otherwise inaccessible and previously unstudied areas such as under the sea ice. Most ambitiously we will anchor a submarine to the seabed within a polynya and leave it over a winter season to collect data, recovering it the following spring. The PICCOLO team will put instruments on seals which will continuously take data as they dive up and down through the water, sending it back to scientists in real-time via satellite communication links. This wealth of novel data will be analysed by the PICCOLO team, using state of the art computer models, to test our ideas about how the whole complex set of physical, chemical and biological processes affects carbon. Conceptually we will follow an imaginary parcel of water through the system looking at processes between the atmosphere and ocean, biological processes in the surface layer, exchanges between the upper and lower ocean and the final fate of the carbon. The PICCOLO hypotheses address the following: (i) Factors controlling the exchange of carbon dioxide between the ocean and atmosphere and the role of ultra-violet light in controlling the concentration of carbon dioxide in seawater; (ii) The role of light, iron and nutrients in how carbon is processed by the plankton in the water; (iii) The mediating processes governing the export of carbon from the upper ocean to depth; (iv) The processes that take the carbon into the deep ocean on the next stage of its global journey.

This proposal is for a Doctoral Training Centre to provide a new generation of engineering leaders in Offshore & Marine Renewable Energy Structures. This is a unique opportunity for two internationally leading Universities to join together to provide an industrially-focussed centre of excellence in this pivotal subject area. The majority of informed and balanced views suggest approximately 180 TWh/year of offshore wind, ~300km of wave farms (19 TWh/year), 1,000 tidal stream turbines (6 TWh/year) and 3 small tidal range schemes (3 TWh/year) are desirable/achievable using David MacKay's UK DECC 2050 Pathways calculator. These together would represent 30% of predicted actual UK electricity demand. This would be a truly enormous renewable energy contribution to the UK electricity supply, given the predicted increase of electricity demand in the transport sector. The inclusion of onshore wind brings this figure closer to 38% of UK electricity by 2050. RenewablesUK predicts Britain has the opportunity to lead the world in developing the emerging marine energy industry with the sector having the potential to employ 10,000 people and generate revenues of nearly £4bn per year by 2020. The large scale development of offshore renewable energy (Wind, Wave and Tidal) represents one of the biggest opportunities for sustainable economic growth in the UK for a generation. The emerging offshore wind sector is however unlike the Oil & Gas industry in that structures are unmanned, fabricated in much larger volumes and the commercial reality is that the sector has to proactively take measures to further reduce CAPEX and OPEX. Support structures need to be structurally optimised and to avail of contemporary and emerging methodologies in structural integrity design and assessment. Current offshore design standards and practices are based on Offshore Oil & Gas experience which relates to unrepresentative target structural reliability, machine and structural loading characteristics and scaling issues particularly with respect to large diameter piled structural systems. To date Universities and the Industry have done a tremendous job to help device developers test and trial different concepts however the challenge now moves to the next stage to ensure these technologies can be manufactured in volume and deployed at the right cost including installation and maintenance over the full design life. This is a proposal to marry together Marine and Offshore Structures expertise with emerging large steel fabrication and welding/joining technologies to ensure graduates from the programme will have the prerequisite knowledge and experience of integrated structural systems to support the developing Offshore and Marine Renewable Energy sector. The Renewable Energy Marine Structures (REMS) Doctoral Centre CDT will embrace the full spectrum of Structural Analysis in the Marine Environment, Materials and Engineering Structural Integrity, Geotechnical Engineering, Foundation Design, Site Investigation, Soil-Structure Interaction, Inspection, Monitoring and NDT through to Environmental Impact and Quantitative Risk and Reliability Analysis so that the UK can lead the world-wide development of a new generation of marine structures and support systems for renewable energy. The Cranfield-Oxford partnership brings together an unrivalled team of internationally leading expertise in the design, manufacture, operation and maintenance of offshore structural systems and together with the industrial partnerships forged as part of this bid promises a truly world-leading centre in Marine Structures for the 21st Century.

The main goal of typing is to prevent the occurrence of execution errors during the running of a program. Milner formalised the idea, showing that ``well-typed programs cannot go wrong''. In practice, type structures provide a fundamental technique of reducing programmer errors. At their strongest, they cover most of the properties of interest to the verification community. A major trend in the development of functional languages is improvement in expressiveness of the underlying type system, e.g., in terms of Dependent Types, Type Classes, Generalised Algebraic Types (GADTs), Dependent Type Classes and Canonical Structures. Milner-style decidable type inference does not always suffice for such extensions (e.g. the principal type may no longer exist), and deciding well-typedness sometimes requires computation additional to compile-time type inference. Implementations of new type inference algorithms include a variety of first-order decision procedures, notably Unification and Logic Programming (LP), Constraint LP, LP embedded into interactive tactics (Coq's eauto), and LP supplemented by rewriting. Recently, a strong claim has been made by Gonthier et al that, for richer type systems, LP-style type inference is more efficient and natural than traditional tactic-driven proof development. A second major trend is parallelism: the absence of side-effects makes it easy to evaluate sub-expressions in parallel. Powerful abstraction mechanisms of function composition and higher-order functions play important roles in parallelisation. Three major parallel languages are Eden (explicit parallelism) Parallel ML (implicit parallelism) and Glasgow parallel Haskell (semi-explicit parallelism). Control parallelism in particular distinguishes functional languages. Type inference and parallelism are rarely considered together in the literature. As type inference becomes more sophisticated and takes a bigger role in the overall program development, sequential type inference is bound to become a bottle-neck for language parallelisation. Our new Coalgebraic Logic Programming (CoALP) offers both extra expressiveness (corecursion) and parallelism in one algorithm. We propose to use CoALP in place of LP tools currently used in type inference. With the mentioned major developments in Corecursion, Parallelism, and Typeful (functional) programming it has become vital for these disjoint communities to combine their efforts: enriched type theories rely more and more on the new generation of LP languages; coalgebraic semantics has become influential in language design; and parallel dialects of languages have huge potential in applying common techniques across the FP/LP programming paradigm. This project is unique in bringing together local and international collaborators working in the three communities. The number of supporters the project has speaks better than words about the timeliness of our agenda. The project will impact on two streams of EPSRC's strategic plan: "Programming Languages and Compilers" and "Verification and Correctness". The project is novel in aspects of Theory (coalgebraic study of (co)recursive computations arising in automated proof-search); Practice (implementation of the new language CoALP and its embedding in type-inference tools); and Methodology (Mixed corecursion and parallelism).