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Expert Comment: How and why mathematics will both underpin and lead the next generation of AI
Professor Peter Grindrod.
Artificial intelligence (AI) has already transformed how we see the world, and how the world sees us. To date, however, most AI systems fall into a small number of familiar categories. Some are analytical engines operating in data-rich environments: for instance, pattern and object recognition, supervised classification, anomaly detection, control systems, and forecasting across images, video, sensor streams, and high-throughput machines. Others are generative, including large language models, media synthesis tools, and conversational agents.
More recently, we have seen the rise of “agentic” AI: systems that can coordinate many components, tools, and subprocesses to pursue set goals. AI can now perform many high-frequency “grunt” tasks and can both increase bandwidth and free up users to spend more time on the “matters that really matter”.
These approaches have delivered impressive results, but they also share defining weaknesses: opacity and implicit biases. Whether we are discussing a neural network trained on millions of images, or a language model orchestrating external actions or procedures, the internal logic of these systems is often difficult to interrogate, explain, or formally trust. As AI moves from making recommendations to supporting decisions, and then to partial autonomy, this opacity becomes a serious concern.
If we want AI systems to exhibit forms of creativity, abstraction, and imagination that remain uniquely human, we will need new mathematical frameworks.
The challenges are now well rehearsed. Are the data adequate and representative? What biases are embedded in training sets or calibration procedures? Can systems be made fair, and who gets to define “fairness”? What subjectivity and blind spots are (ever) acceptable within distinct applications? How vulnerable are they to malicious manipulation? What can we do about hallucinations?
These are not peripheral questions. They strike at the foundations of what it means for an AI system to function reliably in the real world.
It is precisely at this foundational level that mathematics can and must play a leading role. Too often, mathematics is treated as something that can be “bolted on” to AI, as a tool for interpretation, evaluation, or error analysis. This is a profound misunderstanding. Mathematical structure is not an accessory to intelligent systems; it is their scaffolding. Without it, we are left with heuristics, pragmatics, and empiricism alone, powerful but fragile, effective yet difficult to justify when things go wrong.
In addition, mathematics offers deeper concepts and abstractions that may be catalysts for next-generation AI. Mathematics offers something very distinctive: a language for hard provable results, logic, confidence, and performance bounds – some guarantees of behaviour and (foreseen and unforeseen) performance.
Mathematics gives us principled ways to reason about data, uncertainty, and evidence. Through probability, geometry and topology, it helps us understand the structure and shape of data spaces, why certain representations work, where decision boundaries lie, and how small perturbations can lead to large changes in outcome. Through optimisation, numerical analysis, and dynamical systems, it sheds light on convergence, stability, and failure modes in learning algorithms. Through information theory, it clarifies what can, and cannot, be inferred from finite data.
Many of today’s most transformative technologies rest on mathematical ideas that once seemed abstract or esoteric.
Equally important, mathematics underpins explanatory and exploratory AI. It allows us not just to build systems that perform well, but to ask why they perform as they do. Explainability, interpretability, and robustness are not purely engineering add-ons; they are mathematical properties that can be analysed, proven, and stress-tested. In examples of AI spoofing or having vulnerabilities to adversarial attacks, or hallucinations, there is a need to understand how and why these occur, and to define and justify suitable mitigations. The same is true with issues of operational bias emanating from conditioning data sets and methods, as data drift between calibration and operations. This is the difference between post-hoc explanations and models that are interpretable by design.
There is also a forward-looking dimension. As interest grows in neuromorphic and brain-inspired computing, mathematics becomes even more central. If we want AI systems to exhibit forms of creativity, abstraction, and imagination that remain uniquely human, we will need new mathematical frameworks, drawing on areas such as category theory, stochastic processes, non-classical logics, and the mathematics of learning and adaptation. These are not incremental tweaks to existing architectures; they are conceptual shifts.
This is the intellectual space in which the Erlangen AI Hub is operating.
Our ambition is not merely to make current AI methods safer or more efficient, important though that is, but to solidify their foundations. By bringing powerful abstract ideas from across mathematics into direct engagement with real-world AI challenges, we aim to build systems that are more reliable, more controllable, and more transparent.
Crucially, this work is grounded in practice, responding to national priorities. Our partners include the BBC, Ofcom, Capgemini and many large and small companies spanning industries of all sizes and sectors, policy specialists and regulators, funders, and national strategic decision-makers. The goal is not mathematics for its own sake, but mathematics that yields actionable insight, mathematics that changes what AI can responsibly do.
Mathematics will be both the innovator and the disruptor in the next phase of AI. It will move us beyond systems that merely correlate toward systems that reason, adapt, and justify their actions within known limits.
This approach matters profoundly for the UK. If a UK Sovereign AI initiative simply replicates the trajectories of the United States, China, India, or the European Union, it will struggle to distinguish itself. Scale alone is not our comparative advantage. Intellectual leadership can be. By developing genuinely new concepts, methods, and guarantees for AI, rooted in deep mathematics, we can lead rather than follow.
There is precedent for this. Many of today’s most transformative technologies rest on mathematical ideas that once seemed abstract or esoteric. Public-key cryptography, post-quantum cryptography, compressed sensing, and modern control theory all began as mathematical insights before becoming industrial necessities. AI will be no different.
Mathematics will be both the innovator and the disruptor in the next phase of AI. It will move us beyond systems that merely correlate toward systems that reason, adapt, and justify their actions within known limits. It will help us replace blind trust with warranted confidence. And it will enable forms of creativity, not just in generating content, but in solving problems that are rigorous, accountable, and genuinely new.
If we want AI that society can rely on, mathematics must be at its core. That is not a constraint on progress. It is the condition that makes progress sustainable. The Erlangen AI Hub is an asset to Oxford, to its academic, commercial and institutional collaborators; and to the UK, which must succeed within a global, competitive, community.
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Chris Rokos, who studied maths in the city, will donate £190m to Cambridge University to establish the Rokos School of Government.
The 55-year-old investor will provide initial support to the new school of government of £130 million, plus up to a further £60 million. The subsequent funding will be matched by the university.
Mr Rokos is a billionaire hedge fund manager who founded Rokos Capital Management.
He went to a state primary school before receiving a scholarship to Eton.
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“I was fortunate to be given the opportunity of an education which transformed my life, and I would like to give something back to Britain,” he said.
“My hope is that, in time, the influence of the Rokos School of Government across the world becomes an important element of that soft power which has been a great asset to the UK.”
Mr Rokos studied mathematics at the University of Oxford and then worked at Goldman Sachs before co-founding Brevan Howard asset management.
He has previously provided financial support for the scholarships programme at Eton and initiatives linked to Pembroke College Oxford.
The Rokos School of Government will prepare future political leaders for a domestic and international politics, Cambridge said.
The school will be built on undeveloped land in the Cambridge West Innovation District.
It will begin operations in autumn 2026 from temporary facilities until the building is completed.
University of Cambridge vice-chancellor Professor Deborah Prentice said: “Tackling the enormous challenges facing our world requires radical new ways of thinking and approaches to leadership.
“Cambridge, with its strengths across all disciplines and its convening power, is uniquely positioned to drive this innovation.
“Thanks to Chris’ generous support, the Rokos School of Government will become a place where leaders and governments – both current and future – together with experts from across our institution generate the insights and solutions needed to respond to our rapidly changing world.”
A trust will be formed to oversee the financial undertakings made to the school by donors and the university.
Mr Rokos will appoint two people to manage the affairs of the trust, and the university will appoint two.
The university has nominated pro vice-chancellor for research Professor Sir John Aston and pro vice-chancellor for university community and engagement Professor Kamal Munir.
Mr Rokos has nominated former academic and now-lawyer Dr Christos Nifadopoulos and president of Cambridge’s Girton College Dr Elisabeth Kendall.
The Financial Conduct Authority (FCA) says around 12.1 million agreements could be eligible for payouts, with drivers receiving an average of £829 each under a £7.5 billion redress programme.
Lewis described the move as “unprecedented”, warning that people who delay could lose out or face long waits for compensation.
“You want to be in the group that’s complained, not the group where the lender is having to find you,” he said.
Why Martin Lewis says you must act now
Lewis stressed there are three key reasons to submit a complaint as soon as possible:
- Faster payouts: “You’re likely to be paid out far more quickly sometime in 2026 rather than 2027.”
- Missing records risk: “The lender may no longer have your details you will be far better off if you complain.”
- Outdated personal details: “You might have changed name moved address three times so it could be unable to track you.”
He added that millions have already taken action: “We’ve had 3.6 million complaints most people tell me it’s dead easy.”
Who can claim compensation
The scheme covers car finance agreements taken out between April 2007 and November 2024, including:
- PCP Personal Contract Purchase
- HP Hire Purchase
But it excludes leasing agreements and some low commission deals.
The FCA estimates around 35 percent of agreements were mis-sold, often due to hidden commission structures such as discretionary commission arrangements, which were banned in 2021.
FCA chief executive Nikhil Rathi said the scheme aims to balance fairness and speed: “It will put £7.5 billion back into people’s pockets. Now we need everyone to get behind it and ensure millions get their money this year.”
He added: “Payouts should not be delayed any longer, especially as household bills come under greater pressure.”
Why payouts have increased
While the average payout has increased to £829, the overall compensation pot has dropped from earlier estimates due to:
- Fewer eligible claims now estimated at 12.1 million
- Stricter rules on low commission deals
- Assumptions that fewer people will claim
Martin Lewis says the most important step is simple make a complaint even if you are unsure: “The only way you can know if you were mis-sold is to complain.”
He also warned that relying on lenders to contact you could leave you waiting longer or missing out entirely.
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When payouts will happen
- Some payments could begin this year
- Most claims expected to be settled by the end of 2027
- Earlier complaints likely to be processed first
With billions set to be returned to drivers, this is one of the largest consumer compensation schemes ever seen in the UK.
But as Martin Lewis makes clear, those who act first are likely to benefit most: “You want to complain now.”
The bins will be for food and garden waste, paper and card, dry recyclables such as glass, metal and plastics, and general non-recyclable rubbish.
In some areas, paper and card may still be collected with other dry recyclables, reducing the number of bins to three.
Ministers say this will provide different local authorities with the flexibility to deliver services that work best for their communities.
From 31 March 2026, bin collections across England will change.
The goal is good: more recycling, less landfill.
The risk is real: more bins, new rules, new schedules.Simpler in theory. Overwhelming in practice.
This account is here for one thing: less confusion at home.
— Bintime | Bin-day reminders (@Bintimeapp) March 1, 2026
New rules in England mean up to 4 bins in use for households
Circular economy minister Mary Creagh said: “We are ending the bin collections postcode lottery and making it easier for people to recycle wherever they live.
“Simplifying these rules will cut out carbon, clean up our streets, and help bring pride back into our communities.
“We will continue to work hand-in-hand with local areas to deliver these changes and ensure there’s more recycled content in the products we buy.”
The new system is part of the government’s wider efforts to build a circular economy, keeping resources in use longer and reducing waste.
Previously, local authorities set their own rules around bin types and what materials could be collected, leading to a patchwork of different systems across the country.
The government now aims to standardise collections to ensure more high-quality material can be processed domestically for reuse by manufacturers to make new products.
Officials say the changes could also cut carbon emissions by reducing the amount of rubbish that gets burned.
To help councils roll out the new scheme, the government has provided £340 million in funding.
Can you get fined for putting bins out early?
How to check your local bin rules
To support some local authorities with area-specific delivery challenges, the government said additional support will be provided, such as agreed transitional arrangements, allowing a later implementation date.
Households can check how and when the new rules will apply in their area by visiting the government’s website.
Enter your postcode to check the rules for your area.
More than £78 billion has been allocated to councils in England for this financial year, including funding for introducing weekly food waste collections for all households.
The government has introduced an extended producer responsibility scheme, which requires packaging producers to cover the costs of recycling or waste management.
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Deposit return scheme coming to supermarkets in 2027
It also plans to launch a deposit return scheme in 2027.
This will see shoppers pay a small deposit when buying drinks in plastic bottles or metal cans, which they will receive back when returning the empty containers to retailers.
What do you think about the new bin rules in England? Let us know in the comments.
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