Mathematics, with its capacity for generality and abstract reasoning, is a subject that is unique in its ability to penetrate deep within other disciplines, to provide a common language for establishing communication channels between research communities, and in the longevity of its influence. The Isaac Newton Institute (INI) is an international hub for supporting mathematical sciences research of the highest quality and impact. It attracts world leading researchers, in all areas of mathematics and cognate disciplines, who interact through a variety of long and short thematic programmes as well as associated workshops, follow-on meetings and a plethora of one-off events. Based in Cambridge, and benefiting from a bespoke and iconic building as well as many world-leading facilities of Cambridge University, INI is nevertheless an independent forum serving the whole of UK mathematical sciences. INI celebrates its 25th anniversary this year. To the end of 2016 there had been 129 long-term programmes in total, and over 26,000 INI programme and workshop participants including 81 Rothschild Visiting Professors/Fellows, from Wolf Prize winner Vladimir Arnold in 1992 to Dijkstra Prize winning theoretical computer scientist Cynthia Dwork in 2016. Participants have also included 27 Fields Medalists, 13 Nobel Laureates, 12 Abel Prize winners, 25 Wolf Prize winners and over 50 Clay Senior Scholars as well as numerous winners of major prizes in other disciplines. This does not include unregistered participants, who are welcome to drop-in to events for a couple of days at a time. INI gives UK researchers unparalleled opportunities to work with one another and with a critical mass of leading international figures in their field, unencumbered by teaching or administrative duties. It maximizes potential for knowledge exchange and the dissemination of UK research achievements, while exposing UK early career researchers to world leaders in their discipline. A common strategic position of all Research Councils is to emphasise the importance of innovative mathematical and statistical methods to their science and in the training of young researchers. From its inception, INI's programmes and embedded workshops were demonstrably intra or interdisciplinary and conceived to accelerate research impact within the mathematical and sister sciences. Recently INI has broadened its remit to address fundamental questions in the social sciences, medicine etc. It has also concerned itself with the instigation of mechanisms to support diversity and gender equality in the sciences, and to nurture early career researchers so as to enlarge the people pipeline. The Turing Gateway to Mathematics (TGM) was created in 2013 as the knowledge exchange arm of INI. Since then it has brought the mathematical sciences community together with an impressive range of over 700 partners in business, industry, commerce and government. It has a proven set of pathways to impact, and partners with a range of organisations to assist the whole of the mathematical sciences community. Public engagement events are regularly hosted at INI, including its Rothschild Public Seminars. In addition to the 25th Anniversary events being held at the Institute, a highlight of which will be a discussion between Sir Andrew Wiles and his biographer Simon Singh, INI is organising a "road show" across the UK including talks by Keith Moore, Librarian at the Royal Society, and leading British space scientists. INI is committed to the maintenance of a reputation for creativity and mathematical excellence. This will mean continuing to deliver ground-breaking research of the highest international standard, supporting the UK mathematical sciences community both in academe and beyond, and further extending the reach of mathematics into other disciplines and applications via TGM. Throughout it will strive to maintain the culture of creativity and achievement for which it is widely recognised.

Mathematics is a profound intellectual achievement with impact on all aspects of business and society. For centuries, the highest level of mathematics has been seen as an isolated creative activity, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. "Crowdsourcing" pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs that are just too long and complicated for any human to comprehend, using programs designed to verify hardware. Yet these techniques are currently used in stand-alone fashion, lacking integration with each other or with human creativity or fallibility. Social machines are new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity. Our long-term vision is to change mathematics, transforming the reach, pace, and impact of mathematics research, through creating a mathematics social machine: a combination of people, computers, and archives to create and apply mathematics. Thus, for example, an industry researcher wanting to design a network with specific properties could quickly access diverse research skills and research; explore hypotheses; discuss possible solutions; obtain surety of correctness to a desired level; and create new mathematics that individual effort might never imagine or verify. Seamlessly integrated "under the hood" might be a mixture of diverse people and machines, formal and informal approaches, old and new mathematics, experiment and proof. The obstacles to realising the vision are that (i) We do not have a high level understanding of the production of mathematics by people and machines, integrating the current diverse research approaches (ii) There is no shared view among the diverse re- search and user communities of what is and might be possible or desirable The outcome of the fellowship will be a new vision of a mathematics social machine, transforming the reach, pace and impact of mathematics. It will deliver: analysis and experiment to understand current and future production of mathematics as a social machine; designs and prototypes; ownership among academic and industry stakeholders; a roadmap for delivery of the next generation of social machines; and an international team ready to make it a reality.

Mathematics is a profound intellectual achievement with impact on all aspects of business and society. For centuries, the highest level of mathematics has been seen as an isolated creative activity, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. "Crowdsourcing" pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs that are just too long and complicated for any human to comprehend, using programs designed to verify hardware. Yet these techniques are currently used in stand-alone fashion, lacking integration with each other or with human creativity or fallibility. Social machines are new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity. Our long-term vision is to change mathematics, transforming the reach, pace, and impact of mathematics research, through creating a mathematics social machine: a combination of people, computers, and archives to create and apply mathematics. Thus, for example, an industry researcher wanting to design a network with specific properties could quickly access diverse research skills and research; explore hypotheses; discuss possible solutions; obtain surety of correctness to a desired level; and create new mathematics that individual effort might never imagine or verify. Seamlessly integrated "under the hood" might be a mixture of diverse people and machines, formal and informal approaches, old and new mathematics, experiment and proof. The obstacles to realising the vision are that (i) We do not have a high level understanding of the production of mathematics by people and machines, integrating the current diverse research approaches (ii) There is no shared view among the diverse re- search and user communities of what is and might be possible or desirable The outcome of the fellowship will be a new vision of a mathematics social machine, transforming the reach, pace and impact of mathematics. It will deliver: analysis and experiment to understand current and future production of mathematics as a social machine; designs and prototypes; ownership among academic and industry stakeholders; a roadmap for delivery of the next generation of social machines; and an international team ready to make it a reality.