Probabilistic modelling permeates all branches of engineering and science, either in a fundamental way, addressing randomness and uncertainty in physical and economic phenomena, or as a device for the design of stochastic algorithms for data analysis, systems design and optimisation. Probability also provides the theoretical framework which underpins the analysis and design of algorithms in Data Science and Artificial Intelligence. The "CDT in Mathematics of Random Systems" is a new partnership in excellence between the Oxford Mathematical Institute, the Oxford Dept of Statistics, the Dept of Mathematics at Imperial College and multiple industry partners from the healthcare, technology and financial services sectors, whose goal is to establish an internationally leading PhD training centre for probability and its applications in physics, finance, biology and Data Science, providing a national beacon for research and training in stochastic modelling and its applications, reinforcing the UK's position as an international leader in this area and meeting the needs of industry for experts with strong analytical, computing and modelling skills. We bring together two of the worlds' best and foremost research groups in the area of probabilistic modelling, stochastic analysis and their applications -Imperial College and Oxford- to deliver a consolidated training programme in probability, stochastic analysis, stochastic simulation and computational methods and their applications in physics, biology, finance, healthcare and Data Science. Doctoral research of students will focus on the mathematical modelling of complex physical, economic and biological systems where randomness plays a key role, covering mathematical foundations as well as specific applications in collaboration with industry partners. Joint projects with industrial partners across several sectors -technology, finance, healthcare- will be used to sharpen research questions, leverage EPSRC funding and transfer research results to industry. Our vision is to educate the next generation of PhDs with unparalleled, cross-disciplinary expertise, strong analytical and computing skills as well as in-depth understanding of applications, to meet the increasing demand for such experts within the Technology sector, the Financial Service sector, the Healthcare sector, Government and other Service sectors, in partnership with industry partners from these sectors who have committed to co-funding this initiative. ALIGNMENT with EPSRC PRIORITIES This proposal reaches across various areas of pure and applied mathematics and Data Science and addresses the EPSRC Priority areas of (15. Mathematical and Computational Modelling), (22. Pure Mathematics and its Interfaces) ; however, the domain it covers is cross-disciplinary and broader than any of these priority areas taken in isolation. Probabilistic methods and algorithms form the theoretical foundation for the burgeoning area of Data Science and AI, another EPSRC Priority area which we plan to address, in particular through industry partnerships with AI/technology/data science firms. IMPACT By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to - sharpening the UK's research lead in this area and training a new generation of mathematical scientists who can tackle scientific challenges in the modelling of complex, simulation and control of complex random systems in science and industry, and explore the exciting new avenues in mathematical research many of which have been pioneered by researchers in our two partner institutions; - train the next generation of experts able to deploy sophisticated data driven models and algorithms in the technology, finance and healthcare sectors

Partial differential equations (PDEs) are at the heart of many scientific advances. The behaviour of every material object in nature, with time scales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by deterministic and stochastic PDEs or by equations with similar features. The role of PDEs within mathematics (especially nonlinear analysis, geometry, topology, stochastic analysis, numerical analysis, and applied mathematics) and in other sciences (such as physics, chemistry, life sciences, climate modelling/prediction, materials science, engineering, and finance) is fundamental and is becoming increasingly significant. PDEs have consequently become one of the largest and most diverse research fields of present-day mathematics. There is a serious shortage of UK researchers and specialists in the Analysis of PDEs and related areas of Core Mathematics and its Interfaces, both in academia and industry, particularly compared to other G8 nations. More generally, several EPSRC reports and the 2010 International Review of UK Mathematics have drawn attention to the under-representation of analysis in the UK, compared to the rest of the world. It is therefore important that resources are invested in this area to remedy this deficiency. The central aim of the new Centre for Doctoral Training (CDT) is to produce cohorts of highly trained, outstanding mathematicians with deep expertise and interdisciplinary skills in the analysis/applications of PDEs and related areas of Core Mathematics and its Interfaces. A sizeable yearly cohort will allow the CDT to create new training mechanisms so that the students will learn theory, analysis, and applications in a variety of fields in a coherent manner with a natural progression, by-passing a traditionally separate `pure' or `applied' approach to learning. The training will be fundamentally connected to all aspects of PDEs and their analysis/applications which, because of the prevalence of PDEs in science and engineering, impinge on a majority of the EPSRC CDT call priority areas. Oxford is well placed to play a leading role, building on UK strengths in PDEs and their analysis/applications. The Oxford Centre for Nonlinear PDE (OxPDE) was created in 2007, jointly by EPSRC under a major Science & Innovation Award and the University of Oxford by significant matching funding. OxPDE has attracted a number of outstanding researchers in PDEs and Analysis, forming the largest research group that there has ever been in PDEs in the UK. The proposed CDT is based on this core group, along with a multidisciplinary cluster of high quality researchers with PDEs as a core connection spread across the Mathematical Institute and the Departments of Physics, Computer Science, Statistics, and Engineering Science within Oxford. The supervisors in our team have extensive experience of providing a high-quality research training environment for supporting doctoral level education/research. The University of Oxford is committed to the formation of the new CDT and will provide a significant contribution, in particular funding up to 3 students per year. One of the key partners, BNP Paribas, will undertake to fund 2 DPhil students commencing in 2014/15 and sponsor 2-6 internships per year for the CDT students. The CDT will have an international dimension with Partners from leading academic and research institutions in the US, China, France, Germany, Italy, Norway, Russia, and Switzerland; these partners have offered a variety of support for our CDT including attendance at their courses and funded visits by our students who will be equipped with a different research/education culture and will gain additional expertise which is absent in the UK.

The financial services industry is at the forefront of the digital economy, and is crucial to the UK's, and especially London's, continuing social and economic prosperity. State-of-the-art Financial IT, Computational Finance and Financial Engineering (collectively Financial Computing) research is crucial to our international competitiveness in investment banking, investment funds or retail banking. Academically this DTC focuses on financial computing, as distinct from quantitative finance, already well resourced. Banks and funds view PhD students in science and engineering as an increasingly important and largely untapped talent pool; although one regrettably with little knowledge of finance. The Financial Services Skills Council notes that employers are placing increasing importance on high-level analytical skills, as well as their acute shortage, especially in the newly emerging areas that drive sector growth. This centre completely embraces the spirit of the Digital Economy programme. The proposed DTC is inherently multidisciplinary involving UCL Computer Science, one of the largest leading departments in its field in the UK, with LSE Finance and the London Business School; the two leading academic finance centres in the UK. Key to developing the financial services industry in the Digital Economy is the creation of a new cohort of researchers who have a strong research capability in IT and computation, but also understand finance and the needs of the wholesale financial services industry leading to early adoption of new financial information technology research.The research groups and centres that will participate in this DTC include worldclass groups at: UCL, such as the Software Systems Engineering Group and the Centre for Computational Statistics and Machine Learning, at LSE such as Financial Markets Group, and at the London Business School, including the Management Science and Operations and Finance Subject Areas. The total value of active grants currently held by the participating groups and centres exceeds 20 Million Pounds, and the number of currently registered PhD students exceeds 130. Collaborators in Statistics, Economics, Mathematics and Physics supplement the potential Supervisor pool.A great strength of this DTC proposal is our industry partners, which include: Abbey, Barclays, Barclays Capital, BNP Paribas, Credit Suisse, Deutsche Bank, Goldman Sachs, HSBC, Lloyds TSB, Man Investments, Merrill Lynch, Morgan Stanley, Nomura, RBS and Thomson Reuters. Regarding training and supervision, each DTC PhD student will follow a personally tailored programme of postgraduate courses drawn from the partners covering financial IT, networks & communications, HCI, computational finance, financial engineering and business, supplemented by lectures from our industry partners: * A tailored educational programme comprising graduate-level courses from UCL, LSE and LBS. * An academic supervisor (from UCL, LSE or LBS) and an industrial advisor (a partner bank, fund or Reuters), and a programme of research covered by an MOU. * A research project in financial IT, computational finance or financial engineering. * Training in industry software, such as Reuters 3000 Xtra, through UCL's virtual training floor.* A substantial period of industrial placement as agreed between the academic and industrial supervisors.* A short period at a leading foreign academic centre

CENTRE VISION Our vision for the new CDT in Financial Computing and Analytics is to as a national 'beacon' linking PhD & Masters' students, industry and academia in financial computing and analytics. We and our Industry partners are also central to the forthcoming investments in Big Data from EPSRC and ESRC (e.g. Business Datasafe). Its principal objective is to educate the next generation of elite PhDs with unparalleled, cross-disciplinary expertise in applied computing, analytics and financial mathematics, as well as in-depth sector understanding, to meet an increasing demand for their skills within the Financial Service Industry, Government, Retail and other Service sectors. Our existing DTC in Financial Computing is unique (there is no other research & training activity like it in the world) and by placing our PhD students in financial institutions and regulators it has had a major impact on the UK financial sector, as indicated by the Financial Times article (School for QUANTS) and our Letters of Support. The CDT is a new partnership between UCL, LSE and ICL, all providing MRes courses and PhD supervision. NATIONAL IMPORTANCE & GROWING NEED FOR CROSS-DISCIPLINARY SKILLS London is the world's leading international financial centre and the UK financial services industry is the key sector for the UK economy, contributed £124bn to the UK economy, generating a trade surplus of £36bn in 2010 and employing 1 million people. London is also the location for our financial regulators and world-class Retailers. Our Financial and other Service industries are therefore crucial to the UK's, and especially London's, continuing social and economic prosperity. Although we receive over 600 enquiries/applications per annum, and growing, recent reports by McKinsey and Accenture highlight the major and growing skills shortage of (postgrad) IT/data scientists in the USA 22,000 and the UK 4,000. EPSRC PRIORITIES AND RESEARCH The proposed CDT is aligned to EPSRC priorities across a number of Themes, in particular: Data to Knowledge (an ICT Theme priority), Industrially Focussed Mathematical Modelling (Mathematical Sciences) and New Digital Ventures (Digital Economy). The crucially important IT research challenges in just one area, namely the application of software engineering, AI and verification/correctness to algorithms for automated trading, illustrates the enormous research opportunities. IMPACT The current DTC in Financial Computing is acknowledged by the Department of Business Innovation & Skills as having had a major impact on our financial industry partners and on our academic partners. This will continue with the new CDT, impacting Regulators, government, Retailers and analytics companies. * STUDENTS - In 2011 the Centre funded more female PhD students than males, and in 2012 the Centre started 40 new PhD students if we count DTC funded students, students funded by other sources, such as retail and analytics companies, and industry-based part-time students. * ACADEMIA - UCL, LSE and Imperial College have all appointed new faculty in applied financial computing and business analytics; and UCL and ICL have started new Masters programmes. * INDUSTRY - many of the Banks now have established formal PhD programmes, in part due to the current DTC, and proved lecturers to the partners for industry-oriented programmes. * REGULATORS AND GOVERNMENT- we have placed PhD students in the BoE/FSA/PRA/FCA and the Cabinet Office, and as discussed in the Case for Support, we have held individual meetings and workshops with the Regulators (BoE, PRA, FCA) and with new (Retailer) partners (Tesco, BUPA, Unilever) to discuss how we can support them. * SOCIETAL - we encourage and support our PhD students in launching their own start-up, and we provide Masters and Undergraduate students to London-based start-ups, especially in the area called New Finance (e.g. P2P lending, crowdfunding).

Due to the need for climate change action, UK Government has committed to the ambitious task of achieving 'net zero' greenhouse gas emissions by 2050. However, some emissions from farming, aviation and other activities are very difficult to eliminate. So to reach net zero the UK must also directly remove greenhouse gases from the atmosphere amounting to the equivalent of 100 million tonnes of carbon dioxide/year. Of the Greenhouse Gas Removal (GGR) options available, increasing the carbon stored in the UK's 'treescapes' (forests, hedgerows etc.) has the greatest potential, the lowest cost, and can be started immediately. Planting woodlands can store carbon in standing trees, in forest soils and in timber products. For these reasons, the UK is committed to a huge increase in forest cover. However, our understanding of all these processes and how they vary across locations and over time is incomplete. This major programme will gather evidence, address knowledge gaps and allow decision makers to understand the GGR consequences of different planting options. Woodlands can also deliver many other benefits, creating habitats to conserve wild species, enhancing water quality, regulating rainfall and reducing flood risk, and providing recreation (hence the "GGR+" title). GGR+ will examine all the diverse aspects of forestry to identify "The Right Tree in the Right Place". However, it is equally possible to plant the wrong tree in the wrong place. This can result in damage to biodiversity, and even cause some soils to release huge amounts of carbon into the atmosphere. Also, if certain types of agriculture are displaced, there could be higher imports of food from countries that destroy rainforests to increase farm yields. On top of this, climate change means that many risks (forest fires, extreme weather, disease) are changing faster than ever. "The Right Tree in the Right Place" is not a simple proposition - if we are not careful, and don't consider the complexities properly, the UK's net zero tree planting strategy will be poorly designed, and at worst could result in forests that actually increase climate change. However, even understanding the consequences of planting in different locations is not enough to plan the future of the UK's forests. Land is typically privately owned and Government cannot dictate its use. Rather they need to create the conditions and incentives needed for owners to decide to plant trees. Consequently, GGR+ will also undertake the economic research needed to turn science advice into practice. This challenge cannot be addressed by scientists alone, and the GGR+ partners include UK land use policy makers, including all of the Forestry authorities, the Defra teams responsible for forestry, climate and agriculture (who will use GGR+ to plant 30,000ha/year), as well as the Ministry of Defence (which has huge land holdings). From the private and NGO sectors our partners include massive land owners such as Network Rail and the National Trust (who together will fund over 74,000ha of planting based on GGR+ advice), as well as a network of over 1,400 farmers, the timber and building sector and many other stakeholders. Together with our partners, GGR+ will design innovative "decision support tools"; bespoke software allowing users to examine the effects of a tree planting investment or policy in terms of greenhouse gas storage, food production, incomes of those involved, effects on biodiversity, water quality, flooding, recreation etc. Perhaps most revolutionary, this tool will allow users to specify what outcomes they want and then see what planting, policy or investments they need to get those outcomes. This is an exciting, highly interdisciplinary approach to answering the surprisingly challenging question of finding "The Right Tree in the Right Place" and setting the UK on the path to delivering net zero emissions by 2050.