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Much has been written about the Riemann hypothesis, which many consider to be the most important unsolved problem in mathematics today. It concerns the zeros of the Riemann zeta function, which Riemann introduced in his famous 8-page paper from 1859. This talk aims to explain the motivation behind his paper, which only incidentally mentioned the Riemann hypothesis. This story has many sequels. 

David E. Rowe is Professor Emeritus for History of Mathematics and Natural Sciences at Johannes Gutenberg University in Mainz.

Wednesday 28 October, 5 pm, Andrew Wiles Building, Oxford

The University of Oxford has welcomed the first cohort of DPhil students to Oxford × QRT Labs, part of a long-term research initiative between Qube Research & Technologies (QRT) and the universities of Oxford, Cambridge and Imperial College London.

Four DPhil students make up Oxford × QRT Labs’ first cohort, with one researcher based in each of the Mathematical Institute, Department of Statistics, Department of Engineering Science and Department of Computer Science.

The launch also marks the appointment of Professor Blanka Horvath as Director of Oxford × QRT Labs.

Oxford × QRT Labs forms part of a wider research network bringing together early-career researchers, academics and QRT’s research and technology community. The initiative is designed to support fundamental research while creating opportunities for researchers to exchange ideas across disciplines and institutions and to engage with industry and the wider scientific and technological community.

Across Oxford, Cambridge and Imperial, the first QRT Labs cohort comprises 13 early-career researchers working across mathematics, statistics, computer science and engineering. Their research spans areas including AI and its mathematical foundations, intelligent systems, decision-making under uncertainty, scientific and high-performance computing, optimisation, and advanced mathematical and statistical modelling.

Researchers will also have opportunities to connect across the wider QRT Labs network through seminars, events and an annual conference, encouraging exchange between disciplines, institutions and QRT’s research community.

In the last few years, large language models (LLMs) have undergone significant development: in their capabilities, in the variety of their applications, and in their prevalence in our world. They have evolved from relatively fluent text generators to proficient reasoning models with which most people interact on a daily basis.

While this progress has, in part, been driven by scale, data, and compute as well as careful engineering, an additional contributor to this development has been an improved theoretical understanding of the transformer architecture on which these models are based. In particular, the self-attention mechanism distinguishes the transformer from machine learning models which came before it. The mathematical structure of this component has proven considerably richer than might be expected from its relatively simple formulation. There is, therefore, an ongoing programme which seeks to provide insight into the underlying behaviour of LLMs by establishing a firm theoretical understanding of the transformer.

On August 21, 2026, Oxford Mathematics hosted the second edition of the “Workshop on the Mathematics of Transformers”, following a first edition at DESY in Hamburg in September 2025. The workshop aimed to bring together researchers in multi-agent dynamics, optimal transport, and partial differential equations, with the common objective to improve the theoretical understanding of the transformer and the underlying self-attention mechanism.

Close to 40 participants, coming from different institutions, countries and disciplines, gathered to discuss open questions and results that we hope will shape a fundamental understanding of LLMs.

Programme

The workshop featured six invited talks given by,

• Samira Kabri (Helmholtz Imaging, DESY) on Anisotropic interaction in self-attention layers,
• Thomas Jacob Maranzatto (University of Maryland) and Jan Peszek (University of Warsaw) on Diverse dynamical behaviors in deep linear transformers,
• Domènec Ruiz-Balet (University of Barcelona) on Hardmax attention dynamics,
• Albert Alcalde (University of Erlangen–Nuremberg) on Quantifying low-temperature concentration in mean-field transformers,
• Edoardo Calvello (California Institute of Technology) on Transformers for operator learning on probability measures: from theory to scientific workflows,
• William Gibson (University of Oxford) on Uniform scaling limits in AdamW-trained transformers.

Most of the talks were concerned with the dynamics of self-attention at inference. The starting point in this line of research is a setting where tokens converge to consensus, a small number of clusters, or concentrate on lower-dimensional spaces. Several of the results presented investigated this picture in different regimes. Allowing the interaction between tokens to be anisotropic changes the stationary states of the dynamics. Working outside the gradient flow framework, on a lower-dimensional invariant manifold, makes it possible to observe cycling and bifurcations as well as clustering, and connects the dynamics to the Kuramoto model. The hardmax limit of the attention mechanism leads to a different class of PDE problems. A further result showed that at low temperature the token distribution concentrates quickly into a metastable state, and that the metastability of this state depends on a hyperparameter.

Two talks went beyond this setting. One used the description of attention as a map between probability measures as a design principle rather than as a tool for analysis, and applied it to nonlinear data assimilation, where it improved on ensemble Kalman baselines. The second considered the large-depth limit of transformers trained with AdamW, and established convergence of the hidden state dynamics to a forward-backward system of ODEs, with bounds that do not depend on the number of tokens.

Outlook

The response to this workshop, and to its predecessor, has been overwhelmingly positive. What came out of the day was a sharper sense of the open problems: the discussions brought them into clearer focus and left participants with a better sense of which questions are worth pursuing and which tools are likely to reach them. Several exchanges pointed towards new collaborations across institutions. Given the extent of the engagement and curiosity for the subject, we were very happy to find that a third edition of the workshop was a late-night topic of discussion.

Thanks

The workshop was jointly organised by José A. Carrillo, Daniel Kelly, Konstantin Riedl (University of Oxford) and Tim Roith (Technical University of Munich), and kindly supported by G-Research and the Advanced Grant Nonlocal-CPD: “Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization” of the European Research Council Executive Agency (ERC).

Oxford Mathematics is delighted to be a partner in the Martingale Postgraduate Foundation's programme of scholarships for students wishing to pursue postgraduate studies in STEM (Science, Technology, Engineering, and Mathematics) subjects.

Martingale is a  postgraduate foundation that offers full funding and multi-year support for outstanding UK students to complete STEM PhDs and MSc courses at leading UK research universities. Mathematics at Oxford is a core component of Martingale.

Martingale Scholars will receive funding for the full cost of a PhD or MSc course. The scholarship will also include a full cost of living allowance, as well as an additional budget for research activities. The foundation will prioritise students for whom family income has been or would be a barrier to postgraduate education.

In addition to having their tuition fees fully funded by Martingale, Scholars will also receive access to outstanding career development opportunities. This includes attending leadership events and residentials, internships and industry placements, as well as opportunities to contribute to public engagement projects. Scholars will join a community of researchers and a network of leading universities and businesses across the UK.

Find out more about the scholarships, how to apply and the work and aspirations of the Martingale Foundation whose generosity enables these scholarships to be awarded.

Below is a video of our first cohort of Martingale Scholars talking about their experience.

Count me in: how mathematics explains music - Sarah Hart

Wednesday 12 August 2026, 5.00-6.00 p.m., Lecture Theatre 1, Mathematical Institute, Oxford

Oxford Mathematician Graeme Segal has been awarded the Chern Medal in recognition of an outstanding lifelong achievement at the highest level in the field of mathematics. The medal is named in honour of the Chinese-born American mathematician Shiing-Shen Chern and was announced today at the International Congress of Mathematics (ICM) in Philadelphia, USA.

Graeme Segal, for whom Shiing-Shen Chern was a major inspiration, is a geometer and topologist whose profound contributions at different stages of his career have enriched and advanced both mathematics and physics. His early work established foundational results in algebraic topology (particularly homotopy theory) that had a major influence on the subsequent development of that subject. Later, his powerful and elegant work on quantum field theory and string theory was central to the forging of new links between pure mathematics and theoretical physics, elucidating "the role of the concept of space in fundamental physics" as Graeme himself described it.

Graeme was born in Australia and did his undergraduate studies in Sydney, before coming to England to study in Cambridge and Oxford. He was Lowndean Professor of Astronomy and Geometry, and Fellow of St John’s College, Cambridge from 1990 to 1999 and Senior Research Fellow, All Souls College, Oxford from 1999 to 2009 where he is now an Emeritus Fellow. He is a Fellow of the Royal Society and a former President of the London Mathematical Society and was awarded its Polya Prize in 1991. In 2010 he received the Royal Society’s Sylvester Medal for mathematical research.

The King has approved the appointment of James Maynard as the new Regius Professor of Mathematics at the University of Oxford. He will take up the post in the Mathematical Institute on 1 October 2026, and will also be a professorial fellow of Merton College.

This chair is one of three Regius Professorships of Mathematics in the United Kingdom, the others being at St Andrews and Warwick. The Oxford Regius chair was created in 2016 as part of the 90th birthday celebrations for Queen Elizabeth II. James will succeed the inaugural postholder, Andrew Wiles.

James says of his appointment: "I’m delighted to be appointed as the new Regius Professor of Mathematics. Oxford is a fantastic place to do maths, and I feel exceptionally privileged to have spent most of my career here. It sometimes feels like there are exciting ideas bubbling away and breakthroughs happening in every office of the department. Mathematics is a very special subject where pure abstract thought and logic, often driven by mere curiosity, can lead to breakthroughs improving so many aspects of modern life. I’m rather daunted to follow on from the incredible legacy of Andrew Wiles, but I hope this position can be testament to the ongoing importance and increasing role of mathematics in the world."

Jon Chapman, Head of the Mathematical Institute, said: "We are delighted that James has accepted the Regius Professor of Mathematics. His work has led to some of the most significant advances in number theory of recent decades, and he is widely admired not only for the depth and originality of his research but also for his generosity as a colleague and mentor. As a former Oxford graduate student who has built his academic career here, James exemplifies the extraordinary mathematical talent that our community fosters. I can think of no better person to succeed Andrew Wiles in this prestigious role."

Jennifer Payne, Warden of Merton College, added: “The College offers Professor James Maynard our warmest congratulations on his appointment as the new Regius Professor of Mathematics. It is a fitting recognition of his outstanding contribution to mathematics. We look forward to welcoming him into the College.”

About James Maynard

Professor Maynard is Professor of Number Theory at the Mathematical Institute in Oxford. He took his BA and 'Part III' in Mathematics from Queens' College, Cambridge, followed by a DPhil in Oxford under the supervision of Roger Heath-Brown, completed in 2013. He was then a Fellow by Examination at Magdalen College, Oxford from 2013-17. He held a Clay Fellowship at Oxford before his promotion to professor in 2018. He has also held research and visiting positions at Montreal, Berkeley and at the Institute of Advanced Study in Princeton. Since 2018 he has been a Professor of Number Theory at the University of Oxford, and a member of St John's College.

James is known for his influential work in analytic number theory, particularly on the distribution of prime numbers. He uses a blend of ideas from analysis, combinatorics and algebra to try to answer old questions in number theory. As a postdoctoral researcher he developed a new sieve method for detecting primes in bounded length intervals, and settled a long-standing conjecture of Paul Erdős on large gaps between primes. He then worked on Diophantine approximation, and in joint work with D. Koukoulopoulos he settled the Duffin-Schaeffer conjecture and dramatically improved upon the work of Schmidt concerning simultaneous approximation by rationals with square denominator. He has published a series of works on the distribution of primes in residue classes which goes beyond what follows from the Generalised Riemann Hypothesis.

James has received numerous prizes for his work, including the Whitehead Prize from the London Mathematical Society in 2015, the European Mathematical Society Prize in 2016, the Cole Prize in Number Theory from the American Mathematical Society in 2020, and the prestigious Fields Medal from the International Mathematical Union in 2022 for his "contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation". He was elected to the Academia Europaea in 2020, as a Fellow of the Royal Society in 2023 and as a founding Fellow of the Academy of Mathematical Sciences in 2026. James is an avid Arsenal supporter.

Patrick Farrell has been awarded the 2025 Germund Dahlquist Prize by Society for Industrial and Applied Mathematics (SIAM) in recognition of outstanding contributions to the numerical solution of differential equations and numerical methods for scientific computing.

Patrick is a professor in the Numerical Analysis group here in Oxford Mathematics and a Tutorial Fellow at Oriel College, Oxford. For 2025 and 2026 he is also the Donatio Universitatis Carolinæ Chair at the Faculty of Mathematics and Physics at Charles University in Prague.

Patrick works on the numerical solution of partial differential equations, with a particular focus on finite element methods; bifurcation analysis of nonlinear PDE; adjoint techniques, their application and automation; and preconditioners and fast solvers. He has applied the numerical techniques he develops to problems in the areas of mixtures, renewable energy, cardiac electrophysiology, glaciology, magnetohydrodynamics, quantum mechanics, and liquid crystals.

Patrick will deliver a lecture titled, “Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables” at Scientific Computing and Differential Equations 2026 at the end of the month. In July 2026 he will give a section lecture at the International Congress of Mathematicians.

PS: they are running a year behind on the prize because of the Pandemic, hence 2025.