The UK continues to be a world leader in geometry and topology. We aim to support this Research Area by:
- supporting strong leadership within the community of researchers working in geometry and topology, to ensure that existing excellence is preserved in the longer term and that the UK continues to have specialist capabilities to lead the global research landscape
- continuing to regard the supporting and strengthening of links to other areas of the mathematical sciences as a key priority, due to the broad nature of research within the mathematical sciences
- supporting research focused on overarching government priorities including but not limited to data science and robotics and autonomous systems; geometric and topological research has the potential to play a transformative role in data analytics (through topological data analysis, for instance)
- continuing to support the development of emerging research topics and to build upon areas of current UK strength to produce transformative research
- monitoring and supporting the training and development of researchers at all career stages to ensure an adequate balance of new researchers and research leaders
- working with the research community to identify the most appropriate routes to maximise and highlight the impact of ongoing research from this area with respect to the wider scientific community.
The broad base of research within this field allows it to influence and impact in other areas of the mathematical sciences and beyond. Over the last five years, multiple international prizes awarded to UK-based researchers and invitations to speak at the International Congress of Mathematics have continually highlighted this. (Evidence source 1,2)
This is a key fundamental Research Area, interconnected with nearly all subject areas within the Mathematical Sciences theme: e.g. with algebra, number theory, mathematical physics and mathematical analysis. Geometric and topological research also plays an underpinning role in the advances of data science (e.g. topological data analysis) and in robotics and autonomous systems (e.g. through algebraic and differential topology, focusing on the configuration of spaces which are key to robotic mechanisms), with both of these topics identified as strategic priorities by government. (Evidence source 1,2,3,4,5)
The current stature of research in geometry and topology in the UK has led to the UK retaining and attracting world-leading researchers, causing significant growth in areas such as algebraic geometry, geometric analysis and differential geometry. In contrast, research in algebraic topology may be on the decline, in line with the global decrease in activity in this area. This may be due to a re-classification of the subject topic, however, and monitoring of capacity in this area should continue. (Evidence source 1,2,3,4,5)
As with all sub-areas of the mathematical sciences, the academic career path from PhD to established career is a concern. EPSRC currently support a Centre for Doctoral Training (CDT) with direct relevance to geometry and topology which aims to produce future world-leading academic researchers. A current concern for the community is support for early-career researchers and this should continue to be monitored and, where necessary, strengthened to support the UK's leading position in this area. (Evidence source 5)
Contributes to all Outcomes and Ambitions in the Prosperous Nation framework (especially in the field of productivity), with explicit links to Ambitions within the Connected, Resilient and Healthy Nation Outcomes:
C1: Enable a competitive, data-driven economy
C3: Deliver Intelligent Technologies and systems
C5: Design for an inclusive, innovative and confident digital society
Geometry and topology is highly relevant to these Ambitions due to its connection to data science and networks, through but not exclusive to topics such as topological data analysis.
H2: Improve prevention and public health
H3: Optimise diagnosis and treatment
This Research Area can have a long-term impact through development of novel computational and mathematical techniques for prediction, analysis and modelling in healthcare.
R3: Develop better solutions to acute threats: cyber, defence, financial and health
Research from this area will enable development of better solutions to acute threats by underpinning novel techniques related to data analytics.