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Mathematics for Europe Mathematics for Europe The online consultation on mathematics was carried out from 29 January to 15 May 2016 by the European Commission Directorate General for Communications Networks, Content & Technology (DG CONNECT) in the context of a stakeholder consultation to prepare the Horizon 2020 Work Programme 2018-2020. June 2016 Digital Agenda for Europe 2 3 FOREWORD This report presents the results of the online consultation on mathematics1 launched early 2016 to nourish the future Horizon2020 Work Programme (2018-20) with innovative mathematical content. Mathematics is recognized today as essential and indispensable for addressing the major challenges in science, technology and society. Faced with the abundance of data on social, technical, economic, ecological, and technological systems new and sophisticated mathematical tools are required for these data to help us tackle pressing societal challenges and provide us with the necessary technological advantages. Whereas mathematics is not a prerequisite in Horizon2020, the various areas covered by the programme rely on its development and its use; HPC, Big data, Quantum computing just to name a few. Without mathematical tools, future research will be severely hampered. The objective of the consultation was to listen to mathematicians and explore the current and emerging challenges in the field of mathematics. What are the emerging key mathematical areas? What is the potential of mathematics for Horizon2020 and beyond? The response was overwhelming with over 180 contributions covering a large variety of fields, submitted by an impressive set of mathematicians from Europe and the world. The results of the consultation confirm that there is enormous variety of mathematical fields and a huge potential for the European mathematical community. It also confirms the message already conveyed by the consultation and workshop on Mathematics and Digital Science carried out in 2014: Mathematics is at the base of modern science, it is essential for the development of computers, the crunching of data, and the research requiring computing power and exploiting big amounts of data. The results of this consultation will feed into the future Horizon2020 Work Programme. The report will be presented at the ICT Proposers Day in September 2016 to enable proposers of future Horizon2020 Work Programme to meet mathematicians and make them aware of the potential of mathematical competence in Europe. However, one report can do little by itself. The European mathematicians need to be active, to raise awareness of the ground breaking role of mathematics, and to ensure that the new and emerging mathematical fields as well as the new implementation possibilities of existing mathematics are fully exploited for the benefit of the European Science and Innovation. The discussion does not close with this report, it has just started. I would like to express my appreciation and thanks to Anni Hellman for launching the debate by means of the consultation and for writing this instructive report. Thank you to everyone who has contributed and for the compilation of the results. Thierry Van der Pyl 1 https://ec.europa.eu/futurium/en/content/open-consultation-mathematics-horizon-2020-next-work-programme 4 5 Table of content CONSULTATION ON MATHEMATICS 1. EXECUTIVE SUMMARY 6 2. BACKGROUND 8 3. MATHEMATICS IN EUROPE TODAY 9 4. COLLABORATION, CONVERGENCE AND INTERACTION 10 5. CHALLENGES 11 6. MATHEMATICS FOR INDUSTRY AND INNOVATION 12 7. MATHEMATICS FOR HPC 13 8. QUANTUM 14 9. DATA ANALYSIS 14 10. MODELLING AND SIMULATION METHODOLOGIES 15 11. BIOMATHEMATICS 15 12. OTHER MATHEMATICAL AREAS 16 13. SUMMARY 16 6 1. EXECUTIVE SUMMARY As some contributors so eloquently express, mathematics can be seen as a soil in which other sciences, technologies and applications are deeply rooted. If we want good fruit and harvest, the soil must be cared for. The future sciences will need new soil to grow, they will need new mathematics. Current developments in big data analysis and HPC’s mathematical basis have shown that existing areas of mathematics formerly deemed merely theoretical, such as algebra and topology, are now very important for these fields. It is therefore impossible to predict what mathematical areas should be in the focus now to be exploitable in 50 years. The mathematicians will be key in identifying the potential of emerging and existing mathematical fields for, amongst other areas, the development of exascale and quantum computing, analytical and simulation tools for big data and for the modelling needed to meet the future environmental, societal and industrial challenges. Mathematics is a prerequisite for contemporary development in most fields of science. It is essential for both the exploitation of big data and the development of HPC towards exascale and quantum. At the same time, these developments are also enablers for new mathematics. The consultation received 181 responses covering a large variety of mathematical disciplines. It is to be noted that this consultation did not aim at covering all fields of mathematics, nor do the contributions do so. The consultation was a fully open, public consultation addressed to professionals who could contribute on a voluntary basis . Many important areas are covered in the contributions received such as data analysis, topological applications, mathematics for HPC, and MSO (Modelling, simulation and optimisation). Biomathematics is also emerging as a new prominent area. Several contributions also address the challenges of the sizes of European Commission grants and the difficulties in matching them with the mathematical and computer science research needs. The concern is that the present format of European funding will not allow Europe to retain its mathematical talent. The contributions can be roughly categorised as follows into areas: 7 The content of contributions is varied: some concentrated on new and emerging mathematical areas needed for the development of HPC, data or better simulation tools. Among these, we find tropical mathematics (algebra and geometry), algebraic-topological-geometrical methods for data analysis, approximate inference methods, stochastic geometric mechanics, integral biomathics, theory of evolving systems, etc. Others pointed at existing mathematical fields that can be exploited for new solutions, such as topology, optimisation and stochastic processes. Problems requiring mathematics were also suggested: mesoscale modelling, systems biology, CAD/CAE in engineering, etc. Quite importantly, the need for collaboration and increasing convergence between different mathematical disciplines as well as between mathematics and other sciences is emphasised in many contributions. Because the online consultation did not equally reach all mathematical communities, some important areas are missing from the contributions and others are only mentioned. This does not make them less essential or the consultation less noteworthy. The results demonstrate the importance of mathematics as a whole. Undoubtedly fields such as probability, statistics and financial mathematics, which appear with just a few mentions in the contributions, are indispensable, with new challenges requiring new approaches. Blockchain, trading algorithms, actuarial challenges of tackling big data and its implications for needs for inverse probability theory approaches, are certainly important for the society at large. Artificial intelligence, robotics and complexity theory as a whole are also fields that require very high level mathematical input. They have been scarcely mentioned in the contributions but this does not mean that they are less important for Europe and for the European Science and Innovation. 8 The contributions came from many sources, whilst some countries were more active than others. The conclusion is clear: Mathematics is a vital part of the European research and innovation landscape. By playing a bigger role in the Horizon2020 calls as a horizontal, cross cutting competence, mathematics would certainly increase the scalability and the quality of the projects. Contributions received also show that there is a real potential for the FET proactives to exploit relevant mathematical fields. 2. BACKGROUND Mathematics is recognized today as essential and indispensable for addressing the major challenges in science, technology and society. New and sophisticated mathematical tools are required to process the abundance of data on social, technical, economic, ecological, and technological systems, and help tackle pressing societal challenges while enabling the necessary technological advances. The overall objectives of the consultation were to inform the future Horizon2020 Work Programme (2018- 20) with innovative mathematical content, and to understand how mathematics could contribute to European research, especially in the area of High Performance Computing, Big data analysis, and in the development of the Digital Single Market1. Concrete and upstream questions were asked: what are the new areas emerging in mathematics that are definitely worth investing in at European level? Where can we expect to produce ground breaking outcomes for Europe? What can mathematics do in key areas such as HPC and big data analysis? What are the other vital areas of mathematics to consider? Additionally, some areas were identified as a starting point to the discussion: topological data analysis and other potential mathematical methods for big data analysis, mathematics for HPC, exascale and exabyte, quantum
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