Our understanding of the natural world, in terms of both Earth systems and deeper physical realities, depends to a large extent on mathematical modelling. Although this has been valuable in a variety of applications, phenomena such as climate change have vastly increased the complexity of the systems that researchers would like to model. For instance, the ocean-climate-human system probably involves numerous feedbacks that are currently beyond mathematical models, as is the economic quantification of “ecosystem services”.1 This will require innovations in solving mathematical challenges such as finding solutions to complex and chaotic equations of fluid flow, and their integration with models that accurately reflect human behaviour and trends in other species’s movements and characteristics at a number of scales. Using higher-resolution data and more powerful computing resources to build and explore models will also improve our understanding.2 Programmes that ensure more equitable access to data and computational resources will bring much-needed depth and strength to these efforts.
The same is true in fundamental physics and cosmology, where insights about both the birth and the deep-future fate of the universe strongly suggest a need for new mathematical ideas and conceptualisations.3 In universe-modelling, previous generations of mathematicians have had predictive success, but more recently, mathematics has responded to discoveries by physicists. However, both groups now need to think about possible new physics and mathematics that will transcend the ideas of quantum mechanics and space-time that dominated the 20th century. These new ideas will be simple, rather than complex, baroque constructions. Insights into novel geometric structures and their properties are proving promising avenues for advances in fundamental physics,4 and the hope – based on past trends – is that such advances will trickle down to other disciplines and fields, seeding progress across the sciences.
