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In the 21st century, mathematical fields such as linear algebra, differential calculus and statistical analysis underpin a wide swathe of our everyday activities, from online shopping to medical diagnosis. \n\nThe field has always been aware of its shortcomings: mathematics has a long tradition of raising awareness of unsolved challenges. At the beginning of the 20th century, for instance, the mathematician David Hilbert outlined 23 significant challenges, 20 of which now have full or partial solutions. At the turn of this century, the Clay Institute offered a $1 million prize for the solution of any of seven problems, only one of which has been solved so far.\n\nMathematicians are aware of plenty of other less-celebrated gaps that also need to be filled, both for the “completeness” of mathematics and because those gaps can contain tools useful for the advancement of humanity’s interests. It has always been extremely difficult to predict the pace of progress and breakthrough in mathematics, and to know which as yet undeveloped mathematical tools will be useful to academics in other disciplines of the natural and social sciences. However, it is clear from past experience that future developments in a variety of mathematical disciplines will accelerate progress in science and technology, facilitate greater societal stability and push forward the frontiers of medicine.\n\nAchieving these advances will require increased collaboration between mathematicians and academics from other disciplines, and improvements in the equity of access to advanced mathematics education.\n\n\n**KEY TAKEAWAYS**\n\nHumans have been counting, measuring and comparing aspects of the physical world, as well as defining abstract mathematical concepts and their interactions, for many tens of thousands of years. While the uses of mathematics have broadened, some of the central original applications remain, albeit with significantly more complexity. Modern mathematics is essential to the study of **Nature**, whether that is for understanding our planet’s past and possible futures, or broader cosmological considerations. Taking this further will require new mathematical tools. The wealth of data available to us in the 21st century has facilitated the development of **Machines** that can assist with the complexity of modern mathematical calculation and logical inference. With judicious development, such AI could soon become a useful tool for mathematicians. Whether in supply chains, medical technologies, transport logistics or public administration, the central role of mathematics in human **Society** is set to continue. Suitably applied, mathematical models can assist with the development of better futures for human groups of all sizes, though challenges remain. There are also challenges in applying mathematics to the study of biological **Life**, which necessarily involves dealing with complex components and processes that are often difficult to formalise as mathematical entities and procedures. Nonetheless, progress is being made, raising the possibility that the application of mathematics can improve future human health and our scientific understanding of life."},"intro":{"text":"Unlike any other discipline, mathematics is deeply rooted in almost every human culture. This has been the case for as long as written records have existed. The practices of counting, measuring, arranging, quantifying and comparing objects in the physical world around us is so useful that we begin to teach them to our children at a very young age.\n"},"anticipatoryImpact":{"text":""},"indicatorValues":[{"id":"68a693620ac1330579fd7999","value":"0.5677","numericValue":0.5677,"year":2025,"indicator":{"id":"65c55cf29e947c438698aa4d","name":"Anticipation Potential","title":null,"slug":"anticipation-potential","dataSetId":"ANTICIPATION_POTENTIAL","color":null}}],"editions":[{"id":"684951c963371e51d83bdf31","name":"2025","slug":"2025","numericValue":2025}],"anticipatoryImpactImage":{"image":{"id":"image_gesda-platform/image-asset/psp-pl-5-25-5-3_image__PSP-PL5_25_5.3_w1whup","url":"https://res.cloudinary.com/shapeable/image/upload/v1760070849/gesda-platform/image-asset/psp-pl-5-25-5-3_image__PSP-PL5_25_5.3_w1whup.webp"}},"embeds":{"citations":[],"imageAssets":[]},"surveyObservations":{"text":"Mathematics is poised to fundamentally reshape scientific and technological frontiers, with experts anticipating significant breakthroughs across its various domains. Areas where mathematics intersects with **Machines**, particularly AI and robotics, are seen as holding the most transformative potential, with impactful advances already being deployed. While the application of fundamental mathematics to describe the mechanics of **Life** also presents considerable transformative opportunities, there is a higher degree of uncertainty surrounding these future breakthroughs. Developments in mathematics related to **Nature** and **Society** are likewise expected to undergo substantial scientific and technological shifts, albeit over a longer time frame. Across these diverse sub-topics, coordinated international action is frequently highlighted as crucial for exploiting anticipated opportunities."},"color":{"id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A","darkValue":null,"veryDarkValue":null},"banner":{"id":"6a922bcfd215991948805d20","name":"\"Cellulose Planets\" by Elise Ansart, ETH Zurich","description":{"text":"\"Cellulose Planets\" by Elise Ansart, ETH Zurich"},"image":{"id":"image_gesda-platform/banner/cellulose-planets-by-elise-ansart-eth-zurich-1_image__22Cellulose_Planets_22_by_Elise_Ansart_ETH_Zurich_zadu9e","url":"https://res.cloudinary.com/shapeable/image/upload/v1787964352/gesda-platform/banner/cellulose-planets-by-elise-ansart-eth-zurich-1_image__22Cellulose_Planets_22_by_Elise_Ansart_ETH_Zurich_zadu9e.jpg","thumbnails":{"mainBanner":{"url":"https://res.cloudinary.com/shapeable/image/upload/c_limit,w_1440/v1787964352/gesda-platform/banner/cellulose-planets-by-elise-ansart-eth-zurich-1_image__22Cellulose_Planets_22_by_Elise_Ansart_ETH_Zurich_zadu9e.jpg","url2x":"https://res.cloudinary.com/shapeable/image/upload/c_limit,w_2880/v1787964352/gesda-platform/banner/cellulose-planets-by-elise-ansart-eth-zurich-1_image__22Cellulose_Planets_22_by_Elise_Ansart_ETH_Zurich_zadu9e.jpg"}}}},"chartImage":null,"citations":[],"subTopics":[{"id":"68e79218996e064174aade13","name":"Nature","path":"/sub-topics/nature","outlineNumber":"5.3.1","slug":"nature-1","__typename":"Platform_SubTopic","color":{"id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","value":"#D3623A"},"topic":{"id":"68a693650ac1330579fd799f","slug":"mathematics","path":"/topics/mathematics"},"intro":{"text":"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](/citation/2025-05-5-3-1/) 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](/citation/2025-05-5-3-2/) Programmes that ensure more equitable access to data and computational resources will bring much-needed depth and strength to these efforts."},"description":{"text":"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](/citation/2025-05-5-3-3/) 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](/citation/2025-05-5-3-4/) and the hope – based on past trends – is that such advances will trickle down to other disciplines and fields, seeding progress across the sciences."},"anticipationScores":{"text":""},"anticipationScoresImage":{"id":"68e8a24d63d1c853e9788cd5","image":{"id":"image_gesda-platform/image-asset/5-3-1-sub-anti-2026_image__5.3.1_sub_anti_2026_e8ww8m","url":"https://res.cloudinary.com/shapeable/image/upload/v1760076352/gesda-platform/image-asset/5-3-1-sub-anti-2026_image__5.3.1_sub_anti_2026_e8ww8m.webp","url2x":null,"width":1200,"height":1200}},"horizons":[{"__typename":"Platform_Horizon","id":"68e79207996e064174aade0e","name":"25-year horizon","slug":"25-year-horizon-20","intro":{"text":"AI informs bio-abundance predictions"},"description":{"text":"AI enables mathematicians to reliably predict the abundance of ocean species."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89c","name":"25-year horizon","slug":"25-year-horizon","years":25,"title":"25-year","subtitle":"horizon"},"embeds":{"citations":[]}},{"__typename":"Platform_Horizon","id":"68e791e2996e064174aade09","name":"10-year horizon","slug":"10-year-horizon-21","intro":{"text":"Mathematicians quantify value of ecosystem services"},"description":{"text":"Innovations in mathematics allow researchers to quantify the value of ecosystem services such as carbon sequestration. Bioeconomic models are enhanced by real-time data from satellite monitoring and other sources."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89b","name":"10-year horizon","slug":"10-year-horizon","years":10,"title":"10-year","subtitle":"horizon"},"embeds":{"citations":[]}},{"__typename":"Platform_Horizon","id":"68e791ba996e064174aade04","name":"5-year horizon","slug":"5-year-horizon-20","intro":{"text":"Research models effects of policy on humans"},"description":{"text":"Enhanced mathematics for studying feedback loops provides a way to model human effects such as the benefits of actions resulting from the Paris agreement. This stimulates further research and greater confidence in policy-making. Mathematicians provide conceptual tools for quantising gravity where space, time and quantum mechanics are secondary, perhaps emergent, phenomena and not central pillars of the model. Geometric renderings of physical properties of fundamental systems offer insights into ways to simplify representations of particle and other interactions."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89a","name":"5-year horizon","slug":"5-year-horizon","years":5,"title":"5-year","subtitle":"horizon"},"embeds":{"citations":[]}}],"indicatorValues":[{"id":"68edecd8af9e6d6d63271214","value":"0.580","numericValue":0.58,"year":2025,"indicator":{"id":"65c55cf29e947c438698aa4d","name":"Anticipation Potential","title":null,"slug":"anticipation-potential","dataSetId":"ANTICIPATION_POTENTIAL","color":null}}],"embeds":{"citations":[{"slug":"2025-05-5-3-1","url":"https://doi.org/10.1038/s44183-024-00073-7","name":"Ecosystem services “on the move” as a nature-based solution for financing the Global Biodiversity Framework","authors":[{"name":"A. M. M. Sequeira et al."}],"authorShowsEtAl":null,"edition":null,"publication":"npj Ocean Sustain","accessDate":null,"startPage":38,"volume":3,"footnoteNumber":1,"year":null},{"slug":"2025-05-5-3-2","url":"https://doi.org/10.1038/s41558-024-02095-y.","name":"Pushing the frontiers in climate modelling and analysis with machine learning","authors":[{"name":"V. Eyring et al."}],"authorShowsEtAl":null,"edition":null,"publication":"Nat. Clim. Chang.","accessDate":null,"startPage":916,"volume":14,"footnoteNumber":2,"year":null},{"slug":"2025-05-5-3-3","url":"https://arXiv.org:2206.06762","name":"Quantum Gravity in 30 Questions","authors":[{"name":"R. Loll et al."}],"authorShowsEtAl":null,"edition":null,"publication":"Arxiv.org","accessDate":null,"startPage":null,"volume":null,"footnoteNumber":3,"year":null},{"slug":"2025-05-5-3-4","url":"https://arXiv.org:1312.2007","name":"The Amplituhedron","authors":[{"name":"N. Arkani-Hamed and J. Trnka"}],"authorShowsEtAl":null,"edition":null,"publication":"Arxiv.org","accessDate":null,"startPage":null,"volume":null,"footnoteNumber":4,"year":null}],"imageAssets":[]}},{"id":"68e792ff996e064174aade2a","name":"Machines","path":"/sub-topics/machines","outlineNumber":"5.3.2","slug":"machines-1","__typename":"Platform_SubTopic","color":{"id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","value":"#D3623A"},"topic":{"id":"68a693650ac1330579fd799f","slug":"mathematics","path":"/topics/mathematics"},"intro":{"text":"Recent progress in the development of AI and machine learning (ML) has led to numerous scientific, medical and even societal innovations that are set to improve the human experience. However, a range of significant challenges remain.[5](/citation/2025-05-5-3-5/) Mathematicians are still seeking a clear theoretical understanding of exactly how and why AI and ML work,[6](/citation/2025-05-5-3-6/) for example, with some models showing a mysterious ability to solve mathematical problems through guesswork in a manner that has yet to be understood. For this and other reasons, it is likely that AI will be most useful in its performance on tasks that are extremely challenging for human intelligence, and AI and human approaches to mathematics could ultimately be complementary rather than competitive.[7](/citation/2025-05-5-3-7/)"},"description":{"text":"A complication comes from the tendency for AIs doing mathematics to “hallucinate” mathematical truths in ways that make their output unsuitable for use in formal proof. As yet, the field has yet to agree on what constitutes a good set of benchmarks for AI performance in a number of mathematical fields,[8](/citation/2025-05-5-3-8/),[9](/citation/2025-05-5-3-9/) making it extremely difficult to measure progress. Nonetheless, it is expected that theoretical advances in AI and ML will make their mathematics increasingly reliable and useful for pressing issues such as modelling the interactions of ocean, human and climate systems, as well as in basic science.[10](/citation/2025-05-5-3-10/)\n\nGenerative AI and statistical ML are being used by applied mathematicians and assisting with the design of physical systems. However, building AI that understands the real world means building “embodied” AI that works via the maths-based rules behind real-world physical processes, integrating symbolic and physical models. This will also produce more robust, efficient and explainable intelligence. Artificial general intelligence will not be achieved without new mathematical insights and architectures that unify and streamline the variety of paths currently being taken."},"anticipationScores":{"text":""},"anticipationScoresImage":{"id":"68e8968b63d1c853e9788bfa","image":{"id":"image_gesda-platform/image-asset/5-3-2-sub-anti-2026_image__5.3.2_sub_anti_2026_grrwnw","url":"https://res.cloudinary.com/shapeable/image/upload/v1760073343/gesda-platform/image-asset/5-3-2-sub-anti-2026_image__5.3.2_sub_anti_2026_grrwnw.webp","url2x":null,"width":1200,"height":1200}},"horizons":[{"__typename":"Platform_Horizon","id":"68e792f6996e064174aade25","name":"25-year horizon","slug":"25-year-horizon-21","intro":{"text":"AI ubiquitous in mathematics research"},"description":{"text":"AI is integrated into the workflow of most mathematicians’ research as a kind of digital assistant for accomplishing routine tasks. Mathematical innovation allows AI-driven robotics systems to operate safely and intelligently in human environments."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89c","name":"25-year horizon","slug":"25-year-horizon","years":25,"title":"25-year","subtitle":"horizon"},"embeds":{"citations":[]}},{"__typename":"Platform_Horizon","id":"68e792d6996e064174aade20","name":"10-year horizon","slug":"10-year-horizon-22","intro":{"text":"AI assists with proofs"},"description":{"text":"AI fed small lemmas is able to formally prove modular parts of large formal proofs. Projects that data-dump iterations of flawed proofs enable AI to learn the process of human theorem-proving. Success here births a database of mathematical activity that complements mathematical achievement and provides AI with training data for achieving mathematical progress in the same style as human mathematicians. AI assists a number of projects centred on climate change, such as quantifying the value of ecosystem services (carbon sequestration, for example), intergenerational discounting, dynamic ocean modelling that manages climate-related displacement of biomass and integrating bioeconomic models into mainstream economic thought. AI-driven robotics interacts with the physical world and learns to mathematically encode physics through experience."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89b","name":"10-year horizon","slug":"10-year-horizon","years":10,"title":"10-year","subtitle":"horizon"},"embeds":{"citations":[]}},{"__typename":"Platform_Horizon","id":"68e792b2996e064174aade1b","name":"5-year horizon","slug":"5-year-horizon-21","intro":{"text":"Benchmarks help improve AI performance in mathematics"},"description":{"text":"A rolling set of benchmarks for measuring AI progress in solving partial differential equations is established. AI assists formal verification routines to correct errors in proofs. Improved access to data assists ML modelling of feedback between ocean, human and climate systems. AI integrates symbolic and physical models for problem-solving."},"color":{"__typename":"Platform_Color","id":"6a9a0ba358e76338f2a563dc","name":"Terracotta","slug":"terracotta","value":"#D3623A"},"type":{"__typename":"Platform_HorizonType","id":"65c55ce79e947c438698a89a","name":"5-year horizon","slug":"5-year-horizon","years":5,"title":"5-year","subtitle":"horizon"},"embeds":{"citations":[]}}],"indicatorValues":[{"id":"68edecf3af9e6d6d6327121f","value":"0.540","numericValue":0.54,"year":2025,"indicator":{"id":"65c55cf29e947c438698aa4d","name":"Anticipation Potential","title":null,"slug":"anticipation-potential","dataSetId":"ANTICIPATION_POTENTIAL","color":null}}],"embeds":{"citations":[{"slug":"2025-05-5-3-5","url":"https://doi.org/10.1038/s41597-024-03099-1","name":"AI and the democratization of knowledge","authors":[{"name":"C. Dessimoz and P. D. 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Calculus underpins our understanding of economic systems; algebra and game theory define the forces at work when we model social interactions such as elections and responses to policy changes such as inter-state relations and healthcare provision; a deep grasp of statistics is vital to the task of anticipating future societal needs and ensuring that resource management is achieved effectively and efficiently. However, individual human behaviour resists mathematical modelling[11](/citation/2025-05-5-3-11/) because of its complicated nature, including susceptibility to priming and framing effects that exert considerable influence on short timescales."},"description":{"text":"Mathematical tools may nonetheless be able to extract rules and descriptions for human behaviour in the aggregate.[12](/citation/2025-05-5-3-12/) There are, for instance, mathematical relationships between quality of societal infrastructure, population size, crime statistics and income distributions. In addition, understanding the kinds of network structures that exist in online and other communities, or organisational structures in particular disciplines, can help describe and model human behaviour and characteristics. Such phenomenological modelling can reveal the core dynamics that drive large-scale transformations in complex systems where first-principles models are impossible. These insights, if gained with sufficient mathematical rigour, can be applied to help shape human behaviour, not just to describe it. Data analysis carried out on societal systems facilitates the exposure of systemic risks[13](/citation/2025-05-5-3-13/) or hidden biases, such as might be found in legal, governmental or corporate decision-making. Mathematically-derived insights can also provide ways to go beyond simple market dynamics, facilitating new, urgently required hybrid markets such as those needed for sustainable development and healthcare.[14](/citation/2025-05-5-3-14/) Understanding of social-network structures can help with robust communication in an information-saturated world, enabling the exchange of ideas beyond the originator’s bubble or to avoid echo-chamber effects.[15](/citation/2025-05-5-3-15/)\n\nOne impediment to progress is not lack of mathematical tools but a lack of data on human behaviour and decision-making. Historical records are too sparse to create precise economic models, and there is a dearth of controlled experiments generating useful, cleanly interpretable data. 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