Oxford Mathematician Jagoda Kaszowska-Mojsa has won the Excellence in Artificial Intelligence Award for her MACROPRU project at the Perspektywy Women in Tech Summit in Warsaw, Poland.

Jagoda's goal is to investigate how new macroprudential policies can influence financial stability without contributing to inequality in society. In her project she is applying cutting-edge, agent-based simulation, big data and AI techniques to uncover the redistributive effects of public policies and to examine the optimal combination of macroprudential tools from a social welfare perspective.

Jagoda developed an agent-based simulation with an AI component. In other words, she simulated the behaviour of the economy as if we were modelling a virtual reality in which entities make informed decisions, learn and create the reality in which they coexist. This is a major shift in the modelling approach in economics and finance that could spark a paradigm shift.

Oxford Mathematician Jared Duker Lichtman explains his fascination and frustration with a conjecture that has puzzled mathematicians for years.

"The Erdős Primitive Set Conjecture centres around one key definition. A set of positive integers $A\subset \mathbb{Z}_{>1}$ is called primitive if no number in $A$ divides another. For example, the primes form a primitive set. More generally, for any $k\geq 1$ the set of numbers with exactly $k$ prime factors (counted with repetition) is also primitive.

The definition of primitivity is quite simple, so it forms a very broad class of sets and it is easy to construct many examples. An example of particular significance historically is the set of perfect numbers. Since Ancient Greece we say a number is perfect if it equals the sum of its proper divisors. For instance 6 is perfect since its proper divisors 1,2,3 sum to 6 itself. Perfect numbers have fascinated mathematicians for millennia, and it is a nontrivial fact that they form a primitive set.

We number theorists often think of the set of primes as a precious gem like a rare diamond. Similarly, we may think of the broader class of primitive sets like a larger treasure trove of jewels, including emeralds, rubies, and sapphires. Each primitive set has its own special properties, just as each gem has its own unique characteristics, like brilliance, colour, and rarity.

We know that the primes become quite rare further out along the number line. Technically speaking, the primes have density zero.

In 1935, the great mathematician Paul Erdős generalized this result considerably. He proved that every primitive set has (lower) density zero. In fact, he showed the stronger result

Theorem (Erdős, 1935): We have uniformly for all primitive $A$,

$$f(A)=\sum_{a\in A}\frac{1}{a\log a} < \infty.$$

His proof also interpreted these series $f(A)$ as encoding the size of density zero objects $A$. Rather, $f(A)$ roughly measures the density of the multiples generated by $A$.

Once we know these series converge, it is natural to ask for a maximum. In 1988, Erdős famously asked if the primes $\mathcal P$ are maximal among all primitive sets.

Erdős Primitive Set Conjecture (1988): We have $f(A) \leq f(\mathcal P)$ for all primitive $A$.

I immediately fell in love with the problem when I first heard it, and had been thinking about it ever since. Usually it can be difficult to say exactly why we feel the way we do about those dear to us. But in this case, the conjecture articulates precisely how the primes are special in a broader context.

For four years, I was wrestling with the conjecture and tried several approaches. After a while I revisited Erdős' original argument from 1935. Roughly speaking, it was very efficient with sets of numbers that were either prime themselves or had only small prime factors, and one could prove the conjecture in this special case. But if any composite numbers had a large prime factor, the Erdős argument became much cruder.

But then in the winter during lockdown, I realized one could leverage ideas from probability theory: morally, I proved that a primitive set cannot contain too many composite numbers with a large prime factor. Thus we can actually avoid many of the crude cases, thereby exploiting extra efficiency from the Erdős argument. From this key connection to probability, a solution finally emerged.

Theorem (L., 2022): The Erdős Primitive Set Conjecture is true.

After all is said and done, we now see another reason why the prime numbers are indeed special."

You can read Jared's proof here and watch him introduce the Conjecture in the short film below. You can also read a feature on Jared in Quanta magazine and a longer explanation of the work on Numberphile.

A popular social media conjecture is that mathematics consists of a series of clever puzzles presented by a crew of witty magicians.

To test this, we spent the marvellous month of May travelling the Andrew Wiles Building, home to Oxford Mathematics, to find out what mathematicians actually do, and why.

The Rising Talents Programme is designed to provide flexible and practical financial support, alongside tools and wider support, for early career women scientists to pursue their research. Five grants are awarded to outstanding women postdoctoral scientists in the fields of Physical Science, Engineering, Mathematics and Computing, Life Science, and Sustainable Development. These fully flexible Fellowships are each worth £15,000.

Bernadette's work develops techniques in topological data analysis (TDA) to study biological data, in particular dynamical networks and spatial data. Her research can be broadly categorised into three main groups: developing TDA techniques to answer biological questions arising from experimental data; developing novel data science methods based on TDA: and using TDA in combination with mechanistic models to link form and function in biological systems.

In her fellowship she will extend her work to mathematical models of tumour vasculature to enable predictions and investigate links between form and function. She will further develop techniques based on persistent homology to quantify heterogeneity in cancer tissue images and develop novel biomarkers for patient stratification, disease phenotyping, treatment prediction, and treatment scheduling. Ultimately, she hopes to make persistent homology biomarkers standard for cancer diagnosis and prognosis.

Bernadette is a Postdoctoral Research Associate in the Centre for Topological Data Analysis in Oxford. She has degrees in Molecular Medicine and also Mathematics (Major) and German Language and Literature Studies (Minor). She did her PhD in the Mathematical Institute, Oxford, (Lincoln College).

Congratulations to Oxford Mathematician Ehud (Udi) Hrushovski who is the joint winner of this year's Shaw Prize in Mathematical Sciences for his "remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences". He shares the prize with Noga Alon, Professor of Mathematics at Princeton University.

Udi is Merton Professor of Mathematical Logic at the University of Oxford and a Fellow of Merton College, Oxford. He studied in the University of California, Berkeley, and worked in Princeton, Rutgers, MIT and Paris and for twenty five years at the Hebrew University in Jerusalem before coming to Oxford.

Udi's work is concerned with mapping the interactions and interpretations among different mathematical worlds. Guided by the model theory of Robinson, Shelah and Zilber, he investigated mathematical areas including highly symmetric finite structures, differential equations, difference equations and their relations to arithmetic geometry and the Frobenius maps, aspects of additive combinatorics, motivic integration, valued fields and non-archimedean geometry. In some cases, notably approximate subgroups and geometric Mordell-Lang, the metatheory had impact within the field itself, and led to a lasting involvement of model theorists in the area. He also took part in the creation of geometric stability and simplicity theory in finite dimensions, and in establishing the role of definable groups within first order model theory. He has co-authored papers with 45 collaborators and has received a number of awards including the Karp, Erdős and Rothschild prizes and the 2019 Heinz Hopf prize.

The Shaw Prize is an annual award first presented by the Shaw Prize Foundation in 2004. Established in 2002 in Hong Kong it honours living individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence.

The Shaw Prize consists of three annual prizes: Astronomy, Life Science and Medicine, and Mathematical Sciences, each bearing a monetary award of US$1.2 million. This will be the eighteenth year of the awards.

Udi becomes the fifth UK-based mathematician to win the prize. All five (Andrew Wiles, Richard Taylor, Simon Donaldson, Nigel Hitchin being the other four) have held faculty positions at Oxford.

We are delighted to announce that Massimiliano Gubinelli has been appointed the new Wallis Professor of Mathematics in Oxford.

Max did his PhD in Theoretical Physics at the University of Pisa, and held professorships at Paris -Sud and Paris Dauphine before taking his current position as the Hausdorff Chair at the Hausdorff Center for Mathematics in Bonn where he specialises in the analysis of stochastic PDEs and the development and generalisation of the Rough Path theory introduced originally by Terry Lyons, whom he succeeds in the Chair.

In particular Max has generalised rough path theory to a wider class of signals, branched rough paths, and has developed other approaches in order to handle more complex dynamics like those underlying parabolic and hyperbolic PDEs.

Today the UK funding bodies have published the results of the UK’s most recent national research assessment exercise, the Research Excellence Framework (REF) 2021.

Research from the Mathematical Institute and the Department of Statistics in Oxford was submitted together under Unit of Assessment 10. Overall, 78% of our submission was judged to be 4* (the highest score available, given for research quality that is world-leading in terms of originality, significance, and rigour).

In a joint statement, the two heads of department, Mike Giles (Mathematical Institute) and Alison Etheridge (Department of Statistics) said:

"This outstanding result is a testament to the breadth, quality and impact of the research produced by colleagues in our two departments, and the outstanding environment in which they work, supported by our excellent professional services staff. We'd like to thank everyone involved in sustaining Oxford Mathematical Sciences, especially those who worked tirelessly in the preparation of the REF2021 submission."

Among the highlights of the research impact case studies we submitted are:

- the use of homogenisation theory and asymptotic analysis in the mathematical modelling of filtration to improve the effectiveness of filters in both commercial applications and the removal of arsenic in groundwater contamination

- statistical analysis of Covid-19 epidemiological data in the early days of the pandemic, including the statistical design and analysis of REACT studies for the assessment of community transmission

Fernando Alday (pictured left) is an Argentinean Theoretical Physicist and Mathematician, Rouse-Ball Professor of Mathematics and Head of the Mathematical Physics Group in Oxford, and a fellow of Wadham College. He did his undergraduate at Centro Atomico Bariloche, Argentina, and his DPhil at SISSA, Italy, under the supervision of Edi Gava and Kumar Narain. He joined Oxford in 2010, after doing Postdocs at Utrecht University in the Netherlands and at the Institute for Advanced Study in the US.

Fernando is well-known for the development of mathematical tools to understand fundamental questions in Quantum Field Theory and Quantum Gravity. His most important contributions involve surprising dualities among different theories and observables in high energy theoretical physics. One of these dualities relates scattering amplitudes to minimal surfaces/soap bubbles in anti-de-Sitter space, while another, known as the AGT correspondence, relates correlation functions in a two-dimensional theory to the spectrum of four-dimensional gauge theories. More recently, Fernando has been developing mathematical tools to compute string and M-theory amplitudes in curved space-time, a subject still in its infancy.

Alain Goriely is a mathematician with broad interests in mathematical methods, mechanics, sciences, and engineering. He is well known for his contributions to fundamental and applied solid mechanics, and, in particular, for the development of a mathematical theory of biological growth, culminating with his seminal monograph The Mathematics on Mechanics of Biological Growth (2017).

He received his PhD from the University of Brussels in 1994 where he became a lecturer. In 1996, he joined the University of Arizona where he established a research group within the renowned Program of Applied Mathematics. In 2010, he joined the University of Oxford as the inaugural Statutory Professor of Mathematical Modelling and fellow of St. Catherine’s College. He is currently the Director of the Oxford Centre for Industrial and Applied Mathematics (OCIAM).

In addition, Alain enjoys scientific outreach based on problems connected to his research, including tendril perversion in plants, twining plants, umbilical cord knotting, whip cracking, the shape of seashells, brain modelling. He is the author of a Very Short Introduction to Applied Mathematics (2017).

Oxford Mathematics now has 31 Fellows of the Royal Society among its current and retired members: John Ball, Bryan Birch, Martin Bridson, Philip Candelas, Marcus du Sautoy, Artur Ekert, Alison Etheridge, Ian Grant, Ben Green, Roger Heath-Brown, Nigel Hitchin, Ehud Hrushovski, Ioan James, Dominic Joyce, Jon Keating, Frances Kirwan, Terry Lyons, Philip Maini, Vladimir Markovic, Jim Murray, John Ockendon, Roger Penrose, Jonathan Pila, Graeme Segal, Endre Süli, Martin Taylor, Ulrike Tillmann, Nick Trefethen, Andrew Wiles, and Fernando and Alain of course.

The eleventh annual two and a half day conference held alternately in Oxford and Cambridge, and focusing on partial differential equations and analysis, took place this year on 11-13th April in the Mathematical Institute in Oxford.

You can now watch (or dip in to) all 18 talks from a range of speakers, local, national & international, including Oxford Mathematician Catalina Pesce, a third-year DPhil student at the EPSRC Centre for Doctoral Training in Partial Differential Equations: Analysis and Applications. In the image Catalina is taking questions at the end of her talk (and doing a little maths in the process).

The European Research Council today announced the winners of its 2021 Advanced Grants competition and Oxford Mathematician Stuart White was one of four awardees from the University of Oxford for his CSTAR project. Just 14% of applications for grants were successful this year - 253 researchers from across the sciences and humanities received awards out of more than 1,700 proposals. Only nine of those 253 were mathematicians.

Stuart's project focuses on the structure and classification of operator algebras. Classification problems are fundamental in mathematics: how do we decide when two things are the same? Examples include the classification of orientable closed surfaces by their genus (the number of 'holes’). So to decide when two such surfaces are homeomorphic (roughly speaking that one can be deformed into the other), one simply checks if they have the same number of holes. The classification questions at the heart of this proposal are for C*-algebras, which can be viewed as non-commutative versions of topological spaces. These are abstract functional analytic objects, with origins in the mathematical formalism of quantum mechanics. A central theme of Stuart’s research is the transfer of ideas between non-commutative measure theory and topology, using results and techniques from Alain Connes’ classification of von Neumann algebras in the 1970s in the setting of C*-algebras.

Stuart says of his award: "I’m really honoured that this project has been selected for funding by the ERC, showing their commitment to fundamental mathematical research, and I’m really excited about building a world class team of postdoctoral researchers and graduate students in operator algebras in Oxford to deliver the goals of the project."

Stuart White did undergraduate study at Jesus College, Cambridge and a PhD at Edinburgh followed by positions at Glasgow University. In 2019 he came to Oxford where he is Professor of Mathematics and Tutorial Fellow, St John's College. In 2021 he was awarded the Whitehead Prize by the London Mathematical Society and in July 2022 will be an invited speaker at the International Congress of Mathematics.

Funded through the European Union, ERC Advanced Grants are designed to support excellent scientists and scholars in any field at the career stage when they are already established research leaders, with a recognised track record of research achievements. ERC funding is for 5 years for a new research project (though of course built on work started previously). The holding of ERC awards by researchers based at UK institutions is subject to formalisation of the UK’s association to Horizon Europe, which remains the stated priority for the UK government, in line with the Trade and Cooperation Agreement agreed between the UK and the EU in December 2020. In the event that association is not confirmed by the final date for signature of grant agreements then the UK government’s Horizon Europe funding guarantee will apply, with UK awardees receiving equivalent funding via UKRI.