̽��ֱ�� of Cambridge - mathematics

Cambridge Festival Speaker Spotlight: Professor Sarah Hart

zs332 — Mon, 03 Mar 2025 08:28:16 +0000

Professor Sarah Hart is a British mathematician specialising in group theory and author. She is Professor Emerita of Mathematics and Fellow of Birkbeck College ( ̽��ֱ�� of London) and has recently been announced as the Mathematical Association President during 2026-27.

Award-winning broadcaster Hannah Fry joins Cambridge as Professor of the Public Understanding of Mathematics

sc604 — Fri, 22 Nov 2024 00:04:32 +0000

Fry brings outstanding experience to the role of communicating to diverse audiences, including with people not previously interested in maths. She will follow in the footsteps of giants of public engagement with mathematics, including David Spiegelhalter and the late Stephen Hawking as she joins the Department of Applied Mathematics and Theoretical Physics (DAMTP).

“I’m really looking forward to joining the Cambridge community,” said Fry, “to those chance encounters and interactions that end up sparking new ideas and collaborations: it’s so exciting to be in an environment where every single person you speak to is working on something absolutely fascinating.”

Fry won the Christopher Zeeman Medal for promoting mathematics in 2018 and the Royal Society David Attenborough Award in 2024, and is the current President of the Institute of Mathematics and its Applications.

She is currently Professor of the Mathematics of Cities at UCL, where she works with physicists, mathematicians, computer scientists, architects and geographers to study patterns in human behaviour – particularly in an urban setting. Her research applies to a wide range of social problems and questions, from shopping and transport to urban crime, riots and terrorism, and she has applied this research by advising and working alongside governments, police forces, supermarkets and health analysts.

“When you create a mathematical model, it doesn’t really matter how beautifully crafted your equations are, or how accurate your simulations are,” said Fry. “You have to think about how the work you’ve created is going to be seen and perceived by other people and how it’s going to be understood or misunderstood.”

̽��ֱ��new professorship builds on Cambridge’s long track record in sharing maths. DAMTP is also the home of the largest subject-specific outreach and engagement project in the ̽��ֱ�� – the Millennium Mathematics Project (MMP).

Fry says she plans for her work at Cambridge to follow on from Spiegelhalter's extensive public communication work, which she sees as a vital part of the research process.

“Communication is not an optional extra: if you are creating something that is used by, or interacts with members of the public or the world in general, then I think it’s genuinely your moral duty to engage the people affected by it,” she said. “I’d love to build and grow a community around excellence in mathematical communication at Cambridge – so that we’re really researching the best possible methods to communicate with people.”

“Hannah is an outstanding mathematician and researcher, and one of the UK’s best maths communicators,” said Professor Colm-cille Caulfield, Head of DAMTP. “Mathematics affects so many aspects of our everyday lives in interesting and exciting ways, and Hannah will strengthen the excellent work already being done at Cambridge in this area. We in DAMTP and our Faculty of Mathematics colleagues in the Department of Pure Mathematics and Mathematical Statistics are so excited to have her join us.”

Professor Fry announced her appointment at an event yesterday (21 November) organised by the MMP in collaboration with the Newton Gateway to Mathematics at the Isaac Newton Institute in Cambridge. ̽��ֱ��event – Communicating mathematical and data sciences – what does success look like? – explored evidence for effectively communicating mathematical and data science research to policymakers, mainstream media and the wider public.

“Professor Fry is one of the most exciting voices in science and mathematics today,” said Professor Nigel Peake, Head of the School of the Physical Sciences. “Her deep commitment to sharing the excitement of maths with people of all ages and backgrounds, at a time when mathematical literacy has never been so important, will be an enormous benefit to Cambridge, and the UK as a whole.”

Professor Hannah Fry, mathematician, best-selling author, award-winning science presenter and host of popular podcasts and television shows, will join the ̽��ֱ�� of Cambridge as the first Professor of the Public Understanding of Mathematics on 1 January.

Lloyd Mann

Hannah Fry

̽��ֱ��text in this work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Images, including our videos, are Copyright © ̽��ֱ�� of Cambridge and licensors/contributors as identified. All rights reserved. We make our image and video content available in a number of ways – on our main website under its Terms and conditions, and on a range of channels including social media that permit your use and sharing of our content under their respective Terms.

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New open-source platform allows users to evaluate performance of AI-powered chatbots

sc604 — Tue, 04 Jun 2024 10:34:36 +0000

A team of computer scientists, engineers, mathematicians and cognitive scientists, led by the ̽��ֱ�� of Cambridge, developed an open-source evaluation platform called CheckMate, which allows human users to interact with and evaluate the performance of large language models (LLMs).

̽��ֱ��researchers tested CheckMate in an experiment where human participants used three LLMs – InstructGPT, ChatGPT and GPT-4 – as assistants for solving undergraduate-level mathematics problems.

̽��ֱ��team studied how well LLMs can assist participants in solving problems. Despite a generally positive correlation between a chatbot’s correctness and perceived helpfulness, the researchers also found instances where the LLMs were incorrect, but still useful for the participants. However, certain incorrect LLM outputs were thought to be correct by participants. This was most notable in LLMs optimised for chat.

̽��ֱ��researchers suggest models that communicate uncertainty, respond well to user corrections, and can provide a concise rationale for their recommendations, make better assistants. Human users of LLMs should verify their outputs carefully, given their current shortcomings.

̽��ֱ��results, reported in the Proceedings of the National Academy of Sciences (PNAS), could be useful in both informing AI literacy training, and help developers improve LLMs for a wider range of uses.

While LLMs are becoming increasingly powerful, they can also make mistakes and provide incorrect information, which could have negative consequences as these systems become more integrated into our everyday lives.

“LLMs have become wildly popular, and evaluating their performance in a quantitative way is important, but we also need to evaluate how well these systems work with and can support people,” said co-first author Albert Jiang, from Cambridge’s Department of Computer Science and Technology. “We don’t yet have comprehensive ways of evaluating an LLM’s performance when interacting with humans.”

̽��ֱ��standard way to evaluate LLMs relies on static pairs of inputs and outputs, which disregards the interactive nature of chatbots, and how that changes their usefulness in different scenarios. ̽��ֱ��researchers developed CheckMate to help answer these questions, designed for but not limited to applications in mathematics.

“When talking to mathematicians about LLMs, many of them fall into one of two main camps: either they think that LLMs can produce complex mathematical proofs on their own, or that LLMs are incapable of simple arithmetic,” said co-first author Katie Collins from the Department of Engineering. “Of course, the truth is probably somewhere in between, but we wanted to find a way of evaluating which tasks LLMs are suitable for and which they aren’t.”

̽��ֱ��researchers recruited 25 mathematicians, from undergraduate students to senior professors, to interact with three different LLMs (InstructGPT, ChatGPT, and GPT-4) and evaluate their performance using CheckMate. Participants worked through undergraduate-level mathematical theorems with the assistance of an LLM and were asked to rate each individual LLM response for correctness and helpfulness. Participants did not know which LLM they were interacting with.

̽��ֱ��researchers recorded the sorts of questions asked by participants, how participants reacted when they were presented with a fully or partially incorrect answer, whether and how they attempted to correct the LLM, or if they asked for clarification. Participants had varying levels of experience with writing effective prompts for LLMs, and this often affected the quality of responses that the LLMs provided.

An example of an effective prompt is “what is the definition of X” (X being a concept in the problem) as chatbots can be very good at retrieving concepts they know of and explaining it to the user.

“One of the things we found is the surprising fallibility of these models,” said Collins. “Sometimes, these LLMs will be really good at higher-level mathematics, and then they’ll fail at something far simpler. It shows that it’s vital to think carefully about how to use LLMs effectively and appropriately.”

However, like the LLMs, the human participants also made mistakes. ̽��ֱ��researchers asked participants to rate how confident they were in their own ability to solve the problem they were using the LLM for. In cases where the participant was less confident in their own abilities, they were more likely to rate incorrect generations by LLM as correct.

“This kind of gets to a big challenge of evaluating LLMs, because they’re getting so good at generating nice, seemingly correct natural language, that it’s easy to be fooled by their responses,” said Jiang. “It also shows that while human evaluation is useful and important, it’s nuanced, and sometimes it’s wrong. Anyone using an LLM, for any application, should always pay attention to the output and verify it themselves.”

Based on the results from CheckMate, the researchers say that newer generations of LLMs are increasingly able to collaborate helpfully and correctly with human users on undergraduate-level maths problems, as long as the user can assess the correctness of LLM-generated responses. Even if the answers may be memorised and can be found somewhere on the internet, LLMs have the advantage of being flexible in their inputs and outputs over traditional search engines (though should not replace search engines in their current form).

While CheckMate was tested on mathematical problems, the researchers say their platform could be adapted to a wide range of fields. In the future, this type of feedback could be incorporated into the LLMs themselves, although none of the CheckMate feedback from the current study has been fed back into the models.

“These kinds of tools can help the research community to have a better understanding of the strengths and weaknesses of these models,” said Collins. “We wouldn’t use them as tools to solve complex mathematical problems on their own, but they can be useful assistants if the users know how to take advantage of them.”

̽��ֱ��research was supported in part by the Marshall Commission, the Cambridge Trust, Peterhouse, Cambridge, ̽��ֱ��Alan Turing Institute, the European Research Council, and the Engineering and Physical Sciences Research Council (EPSRC), part of UK Research and Innovation (UKRI).

Reference:
Katherine M Collins, Albert Q Jiang, et al. ‘Evaluating Language Models for Mathematics through Interactions.’ Proceedings of the National Academy of Sciences (2024). DOI: 10.1073/pnas.2318124121

Researchers have developed a platform for the interactive evaluation of AI-powered chatbots such as ChatGPT.

Anyone using an LLM, for any application, should always pay attention to the output and verify it themselves

Albert Jiang

da-kuk via Getty Images

Chatbot

Yes

New Cambridge-developed resources help students learn how maths can help tackle infectious diseases

sc604 — Mon, 19 Feb 2024 07:00:00 +0000

From measles and flu to SARS and COVID, mathematicians help us understand and predict the epidemics that can spread through our communities, and to help us look at strategies that we may be able to use to contain them.

̽��ֱ��project, called Contagious Maths, was led by Professor Julia Gog from Cambridge’s Department of Applied Mathematics and Theoretical Physics (DAMTP), and was supported by a Rosalind Franklin Award from the Royal Society.

̽��ֱ��curriculum-linked resources will give students between the ages 11 and 14 the opportunity to join researchers on the mathematical frontline to learn more about infectious disease spread, along with interactive tools to try mathematical modelling for themselves. Teachers receive full lesson plans, backed up by Cambridge research.

“I’ve always loved maths. I was lucky enough to have amazing teachers at sixth form who challenged me and were 100% behind me pursuing maths at the highest level, but maths as it’s taught in school can be highly abstract, so students often wonder what the point of maths even is,” said Gog, who is also Director of the Millennium Maths Project. “This is something I’m trying to help with now: to offer a glimpse from school to the research world to see the role mathematics can play in tackling important real-world problems.”

̽��ֱ��Contagious Maths project introduces mathematical modelling; explores how mathematicians can model the spread of disease through a population and the type of questions we might think about when looking at models; and gives an insight into what mathematics researchers working on these real-life problems actually do.

“I’ve been engaged in outreach for many years at Cambridge, and the Contagious Maths project grew out of discussions with colleagues who have expertise in reaching school-age children,” said Gog. “ ̽��ֱ��11-14 age group we are targeting is a real crunch point for retaining girls in maths, and future female mathematicians. What exactly happens is complex and multifaceted, but this is a period when people form their views on how they fit with maths and science.

“Many of them disengage, as it can seem that maths at school is utterly disconnected from the real world. It can also be a time when maths appears very starkly right or wrong, whereas any research mathematician can tell you it’s always so much more subtle than that, and therefore so much more interesting!”

Gog hopes the Contagious Maths resources might be able to help, as they are designed to be used in regular school lessons, and cover a topic with clear real-world importance.

“ ̽��ֱ��maths is never black and white in this field: there are always ways to challenge and develop the models, and some tricky thinking to be done about how the real epidemics and the simulations are really related to each other,” she said. “I suspect some students will find this frustrating, and just want maths to be algorithmic exercises. But some will be intrigued, and they are the ones we are trying to reach and expose to this larger world of applied maths research.”

Contagious Maths also provides teachers with all the ideas and tools they need, so they have at their fingertips all they need to deliver these lessons, even if they have no experience with research mathematics. “We hope this project will help these teachers to bring in the wider view of mathematics, and we hope it inspires them too,” said Gog. “It’s been really fun developing these resources, teaming up with both NRICH and Plus to make the most of our combined expertise.”

Maths teachers can attend a free online event on 20 March to learn more about the project.

In addition to the school resources, Gog and her colleagues have designed another version of Contagious Maths for a more general self-guided audience, which will work for students older than 14 or anyone, of any age, who is interested in learning about mathematical modelling.

“ ̽��ֱ��paradox between the cleanness and precision of mathematics, and the utter hot mess of anything that involves biological dynamics across populations – like an outbreak of an infectious disease, is what intrigued me to stay in mathematics beyond my degree, and to move into research in mathematical biology,” said Gog. “Elegant theoretical ideas can tell us something valuable and universal about mitigating the devastating effects of disease on human and animal populations. Super abstract equations can hold fundamental truths about real-world problems - I don't think I will ever tire of thinking about that.”

Adapted from a Royal Society interview with Professor Julia Gog.

Cambridge mathematicians have developed a set of resources for students and teachers that will help them understand how maths can help tackle infectious diseases.

Orbon Alija via Getty Images

Aerial view of crowd connected by lines

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Swarming cicadas, stock traders, and the wisdom of the crowd

sc604 — Thu, 01 Feb 2024 14:36:51 +0000

Pick almost any location in the eastern United States – say, Columbus Ohio. Every 13 or 17 years, as the soil warms in springtime, vast swarms of cicadas emerge from their underground burrows singing their deafening song, take flight and mate, producing offspring for the next cycle.

This noisy phenomenon repeats all over the eastern and southeastern US as 17 distinct broods emerge in staggered years. In spring 2024, billions of cicadas are expected as two different broods – one that appears every 13 years and another that appears every 17 years – emerge simultaneously.

Previous research has suggested that cicadas emerge once the soil temperature reaches 18°C, but even within a small geographical area, differences in sun exposure, foliage cover or humidity can lead to variations in temperature.

Now, in a paper published in the journal Physical Review E, researchers from the ̽��ֱ�� of Cambridge have discovered how such synchronous cicada swarms can emerge despite these temperature differences.

̽��ֱ��researchers developed a mathematical model for decision-making in an environment with variations in temperature and found that communication between cicada nymphs allows the group to come to a consensus about the local average temperature that then leads to large-scale swarms. ̽��ֱ��model is closely related to one that has been used to describe ‘avalanches’ in decision-making like those among stock market traders, leading to crashes.

Mathematicians have been captivated by the appearance of 17- and 13-year cycles in various species of cicadas, and have previously developed mathematical models that showed how the appearance of such large prime numbers is a consequence of evolutionary pressures to avoid predation. However, the mechanism by which swarms emerge coherently in a given year has not been understood.

In developing their model, the Cambridge team was inspired by previous research on decision-making that represents each member of a group by a ‘spin’ like that in a magnet, but instead of pointing up or down, the two states represent the decision to ‘remain’ or ‘emerge’.

̽��ֱ��local temperature experienced by the cicadas is then like a magnetic field that tends to align the spins and varies slowly from place to place on the scale of hundreds of metres, from sunny hilltops to shaded valleys in a forest. Communication between nearby nymphs is represented by an interaction between the spins that leads to local agreement of neighbours.

̽��ֱ��researchers showed that in the presence of such interactions the swarms are large and space-filling, involving every member of the population in a range of local temperature environments, unlike the case without communication in which every nymph is on its own, responding to every subtle variation in microclimate.

̽��ֱ��research was carried out Professor Raymond E Goldstein, the Alan Turing Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics (DAMTP), Professor Robert L Jack of DAMTP and the Yusuf Hamied Department of Chemistry, and Dr Adriana I Pesci, a Senior Research Associate in DAMTP.

“As an applied mathematician, there is nothing more interesting than finding a model capable of explaining the behaviour of living beings, even in the simplest of cases,” said Pesci.

̽��ֱ��researchers say that while their model does not require any particular means of communication between underground nymphs, acoustical signalling is a likely candidate, given the ear-splitting sounds that the swarms make once they emerge from underground.

̽��ֱ��researchers hope that their conjecture regarding the role of communication will stimulate field research to test the hypothesis.

“If our conjecture that communication between nymphs plays a role in swarm emergence is confirmed, it would provide a striking example of how Darwinian evolution can act for the benefit of the group, not just the individual,” said Goldstein.

This work was supported in part by the Complex Physical Systems Fund.

Reference:
R E Goldstein, R L Jack, and A I Pesci. ‘How Cicadas Emerge Together: Thermophysical Aspects of their Collective Decision-Making.’ Physical Review E (2024). DOI: 10.1103/PhysRevE.109.L022401

̽��ֱ��springtime emergence of vast swarms of cicadas can be explained by a mathematical model of collective decision-making with similarities to models describing stock market crashes.

Ed Reschke via Getty Images

Adult Periodical Cicada

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̽��ֱ��Vice-Chancellor's Awards 2023 for Research Impact and Engagement

zs332 — Wed, 13 Dec 2023 09:20:46 +0000

Meet the winner of the Vice-Chancellor's Awards 2023 for Research Impact and Engagement and learn more about their projects.

NRICH: nurturing next-generation problem solvers

ta385 — Wed, 30 Mar 2022 06:00:00 +0000

NRICH spent the last two years in emergency rescue mode, helping learners in lockdown. Its online resources attracted over a million page views per week. Now celebrating their 25th anniversary, the NRICH team is more determined than ever to nurture our next-generation problem solvers.

Mathematical paradox demonstrates the limits of AI

sc604 — Thu, 17 Mar 2022 16:05:06 +0000

Like some people, AI systems often have a degree of confidence that far exceeds their actual abilities. And like an overconfident person, many AI systems don’t know when they’re making mistakes. Sometimes it’s even more difficult for an AI system to realise when it’s making a mistake than to produce a correct result.

Researchers from the ̽��ֱ�� of Cambridge and the ̽��ֱ�� of Oslo say that instability is the Achilles’ heel of modern AI and that a mathematical paradox shows AI’s limitations. Neural networks, the state-of-the-art tool in AI, roughly mimic the links between neurons in the brain. ̽��ֱ��researchers show that there are problems where stable and accurate neural networks exist, yet no algorithm can produce such a network. Only in specific cases can algorithms compute stable and accurate neural networks.

̽��ֱ��researchers propose a classification theory describing when neural networks can be trained to provide a trustworthy AI system under certain specific conditions. Their results are reported in the Proceedings of the National Academy of Sciences.

Deep learning, the leading AI technology for pattern recognition, has been the subject of numerous breathless headlines. Examples include diagnosing disease more accurately than physicians or preventing road accidents through autonomous driving. However, many deep learning systems are untrustworthy and easy to fool.

“Many AI systems are unstable, and it’s becoming a major liability, especially as they are increasingly used in high-risk areas such as disease diagnosis or autonomous vehicles,” said co-author Professor Anders Hansen from Cambridge’s Department of Applied Mathematics and Theoretical Physics. “If AI systems are used in areas where they can do real harm if they go wrong, trust in those systems has got to be the top priority.”

̽��ֱ��paradox identified by the researchers traces back to two 20th century mathematical giants: Alan Turing and Kurt Gödel. At the beginning of the 20th century, mathematicians attempted to justify mathematics as the ultimate consistent language of science. However, Turing and Gödel showed a paradox at the heart of mathematics: it is impossible to prove whether certain mathematical statements are true or false, and some computational problems cannot be tackled with algorithms. And, whenever a mathematical system is rich enough to describe the arithmetic we learn at school, it cannot prove its own consistency.

Decades later, the mathematician Steve Smale proposed a list of 18 unsolved mathematical problems for the 21st century. ̽��ֱ��18th problem concerned the limits of intelligence for both humans and machines.

“ ̽��ֱ��paradox first identified by Turing and Gödel has now been brought forward into the world of AI by Smale and others,” said co-author Dr Matthew Colbrook from the Department of Applied Mathematics and Theoretical Physics. “There are fundamental limits inherent in mathematics and, similarly, AI algorithms can’t exist for certain problems.”

̽��ֱ��researchers say that, because of this paradox, there are cases where good neural networks can exist, yet an inherently trustworthy one cannot be built. “No matter how accurate your data is, you can never get the perfect information to build the required neural network,” said co-author Dr Vegard Antun from the ̽��ֱ�� of Oslo.

̽��ֱ��impossibility of computing the good existing neural network is also true regardless of the amount of training data. No matter how much data an algorithm can access, it will not produce the desired network. “This is similar to Turing’s argument: there are computational problems that cannot be solved regardless of computing power and runtime,” said Hansen.

̽��ֱ��researchers say that not all AI is inherently flawed, but it’s only reliable in specific areas, using specific methods. “ ̽��ֱ��issue is with areas where you need a guarantee, because many AI systems are a black box,” said Colbrook. “It’s completely fine in some situations for an AI to make mistakes, but it needs to be honest about it. And that’s not what we’re seeing for many systems – there’s no way of knowing when they’re more confident or less confident about a decision.”

“Currently, AI systems can sometimes have a touch of guesswork to them,” said Hansen.“You try something, and if it doesn’t work, you add more stuff, hoping it works. At some point, you’ll get tired of not getting what you want, and you’ll try a different method. It’s important to understand the limitations of different approaches. We are at the stage where the practical successes of AI are far ahead of theory and understanding. A program on understanding the foundations of AI computing is needed to bridge this gap.”

“When 20th-century mathematicians identified different paradoxes, they didn’t stop studying mathematics. They just had to find new paths, because they understood the limitations,” said Colbrook. “For AI, it may be a case of changing paths or developing new ones to build systems that can solve problems in a trustworthy and transparent way, while understanding their limitations.”

̽��ֱ��next stage for the researchers is to combine approximation theory, numerical analysis and foundations of computations to determine which neural networks can be computed by algorithms, and which can be made stable and trustworthy. Just as the paradoxes on the limitations of mathematics and computers identified by Gödel and Turing led to rich foundation theories — describing both the limitations and the possibilities of mathematics and computations — perhaps a similar foundations theory may blossom in AI.

Matthew Colbrook is a Junior Research Fellow at Trinity College, Cambridge. Anders Hansen is a Fellow at Peterhouse, Cambridge. ̽��ֱ��research was supported in part by the Royal Society.

Reference:
Matthew J Colbrook, Vegard Antun, and Anders C Hansen. ‘ ̽��ֱ��difficulty of computing stable and accurate neural networks – On the barriers of deep learning and Smale’s 18th problem.’ Proceedings of the National Academy of Sciences (2022). DOI: 10.1073/pnas.2107151119

Humans are usually pretty good at recognising when they get things wrong, but artificial intelligence systems are not. According to a new study, AI generally suffers from inherent limitations due to a century-old mathematical paradox.

There are fundamental limits inherent in mathematics and, similarly, AI algorithms can’t exist for certain problems

Matthew Colbrook

Yuichiro Chino

Binary data wave

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. Images, including our videos, are Copyright © ̽��ֱ�� of Cambridge and licensors/contributors as identified. All rights reserved. We make our image and video content available in a number of ways – as here, on our main website under its Terms and conditions, and on a range of channels including social media that permit your use and sharing of our content under their respective Terms.

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