Capacity building in Africa via technology-driven research in algebraic and arithmetic geometry Our proposed project aims to build mathematics research capacity on the African continent by establishing deep, meaningful collaborations between UK mathematicians and emerging Sub-Saharan African scientists and their research groups to tackle a range of problems arising in technological challenges relevant to the African development context. Our approach to these problems will involve the toolkit of abstract mathematics. Our objective is to build up expertise among African partners and their research groups in using theoretical approaches to attack applied problems.  Our capacity building programme is focused around collaborative research in applied algebraic and arithmetic geometry. This is a fast-developing area that aims to apply the sophisticated tools of abstract geometry to the study of systems of polynomial equations arising in applications. The specific research objectives we will tackle come from the areas of power flows, systems biology, robotics and cryptography. All these lines of inquiry can address specific challenges arising in development-related contexts. Controlling and understanding power flow in electrical networks leads to improvements in the robustness and hence the reliability of an electrical system without having to resort to expensive precautionary over-engineering. A better theoretical understanding of signalling pathways connected to specific diseases enables more efficient experimental design, with a high potential for applications in an African context. Theoretical progress in robotics leads to suggestions for better practical design of specific kinematical constructions, that can then be tested in emerging local robotics labs. Cryptography, including more sophisticated versions such as elliptic curve cryptography, allows for efficient protocols for information security, and is widely used in the banking sector including mobile money transfers, an industry in which Africa is a world leader.  On the UK side, our capacity building research programme will be led by a group of Investigators with a track record in high-level research in the fields of algebraic and arithmetic geometry and their applications, and experience of research development in an international context. On the African side, we will work with a core group of African Partners, emerging leaders already known to us and active in research in the fields of algebra and algebraic and arithmetic geometry. We will recruit further African participants via open calls to join our capacity building research programme.  Our proposed project will consist of two phases. In the first phase of the project, two intensive, 10-day workshops will be held in East and West Africa, respectively. In the second phase of the project, we will host about African scientists in the UK for visits to the UK, whereas Investigators will visit African partner departments.  Our capacity building programme will allow African mathematicians to find the time and focus to embark on research in specific development-related areas, which are mostly novel for them. Via the impact on their research groups, this will lead to a sustainable increase in local expertise in these lines of inquiry. We will thus build crucial research capacity able to tackle challenges relevant in the context of development. The Global Challenges Research Fund (GCRF) supports cutting-edge research to address challenges faced by developing countries. The fund addresses the UN sustainable development goals. It aims to maximise the impact of research and innovation to improve lives and opportunity in the developing world.