Mon, 27 Jun 2022

All we ever wanted was everything / 24.02.22 (for Ukraine)

Image of sculpture

On June 27th, in the Reception area of the Mathematical Institute, Oxford artist Andy Bullock unveiled his most ambitious knot sculpture to date, a large floor-based work titled ‘All we ever wanted was everything / 24.02.22 (for Ukraine)’ constructed using 70 metres of metal trunking. As with all his knot sculptures they often reference issues of complexity with situations and people, the personal and interpersonal; focusing on what it means to be human.

In a first for the artist, Bullock will be inviting members of the recently arrived Ukrainian refugee community to contribute to the artwork by incorporating items of personal relevance. Bullock is reaching out to Oxfordshire’s Ukrainian community in a collaboration with Yulia Astasheva, a recent arrival herself from the Dnipropetrovsk region, where she still has close family living only miles from the Russian-occupied region.

The idea for the work came initially from a commission from Oxford Mathematics for Bullock to create an exhibition of his maths-related painting, photography and sculpture to be open to the public this summer. The core of his fine art master’s degree show last year was a creative examination and exploration of the topological subject of knot theory, and in particular the work of Clifford Hugh Dowker (1912-82), an eminent mathematician whose work is still studied today. “I find a poetic beauty in the mathematics I researched even though my understanding of the subject is virtually nil” said Bullock. “My final dissertation for my master’s degree examined the similarities in thought of mathematicians working in these areas and that of artists working in a more conceptual arena”.

In the lower ground floor space of the building there is an exhibition of some of Andy Bullock’s ‘knot variation’ paintings and photographs and a display of original handwritten manuscripts from Dowker’s personal archive alongside Andy's own sketchbooks, allowing an insight into the respective processes of mathematician and artist.

The exhibition will run until 22 July.

For further information:

Andy Bullock - @email - 07582 526957 -

Yulia Astasheva - @email

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Thu, 16 Jun 2022

3 Minute Thesis Competition

Image of Benedikt Stock giving his talk; and his slide which features David Hilbert, Yuri Matiyasevich and a cartoon of Benedikt himself.

For 4 years our DPhil (PhD) students go deep into their area of research. Then we suggest they sum it all up in 1 slide and 3 minutes.

6 of them have done just that in our 3 Minute Thesis Competition. Easy-peasy.

Alexander Van-Brunt - PDE theory and the energy storage problem
Sophie Abrahams - How bubbles affect kidney stones
Benedikt Stock - (Un)Decidability in Number Theory
Matthew Cotton - Curvature inducing membrane-bound proteins
Michael Negus - Making an impact with droplet modelling
Jared Duker Lichtman - The Erdos primitive set conjecture

The competition is organised by the Oxford Mathematics SIAM-IMA Student Chapter.

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Mon, 13 Jun 2022

Jagoda Kaszowska-Mojsa wins Excellence in Artificial Intelligence Award

Image of Jagoda with the prize

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.

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Mon, 06 Jun 2022

Oxford graduate student proves decades old Erdős conjecture on primes

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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.

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Fri, 03 Jun 2022

Me and My Maths

Image of Sam Palmer on the  roof terrace of the Andrew Wiles Building

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.

In Episode 1 meet Jacobus, Maria and Sam.

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Wed, 25 May 2022

Bernadette Stolz receives L’Oréal-UNESCO Women in Science Rising Talent fellowship

Bernadette writing on whiteboard

Bernadette Stolz has received a L'Oréal-UNESCO For Women in Science UK and Ireland Rising Talent fellowship in the category of Mathematics and Computer Science.

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).

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Tue, 24 May 2022

Ehud Hrushovski joint winner of this year's Shaw Prize in Mathematical Sciences

Photo of Udi

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. 

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Fri, 13 May 2022

New Wallis Professor of Mathematics

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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.

We look forward to his arrival in Oxford.

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Thu, 12 May 2022

Research Excellence Framework (REF) 2021

Image of graduate students in the Common Room

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 rough path theory to improve the effectiveness of machine learning in Chinese handwritten character recognition for mobile phones

- 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

Full REF results and full University of Oxford results

Photograph by John Cairns

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Tue, 10 May 2022

Fernando Alday and Alain Goriely elected Fellows of the Royal Society

Photos of Fernando and Alain

Today Oxford Mathematicians Fernando Alday and Alain Goriely have been elected Fellows of the Royal Society (FRS) for their outstanding contributions to science.

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.

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