Case Studies and Films

Modelling cylinder addition in Søderberg electrodes

The 7m length by 2m diameter cylindrical Søderberg electrode is the secret behind a yearly production capacity of 215,000 tonnes of silicon for Elkem ASA, the third largest silicon producer in the world. This electrode operates continuously thanks to its raw material: carbon paste, whose viscosity depends very sensitively on the temperature.

Electric arcs and silicon furnaces - a case study by Ellen Luckins

Silicon is produced industrially in a submerged arc furnace (illustrated in Figure 1, left), with the heat required for the endothermic chemical reaction provided by an electric current. The high temperatures within the furnace (up to around 2000 K) prohibit observation of the internal conditions, so that mathematical modelling is a valuable tool to understand the furnace processes.

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumours

A collaboration between pure and applied mathematicians, pathologists and clinicians, has shown how sophisticated techniques from computational algebraic topology can be applied to understand the spatial distributions of immune cell subtypes in the tumour microenvironment.

Forget me not: what maths can tell us about Alzheimer's disease - Travis Thompson

Alois Alzheimer called Alzheimer's disease (AD) the disease of forgetfulness in a 1906 lecture that would later mark its discovery. Alzheimer noticed the presence of aggregated protein plaques, made up of misfolded variants of amyloid-beta (A$\beta$) and tau ($\tau$P) proteins, in the brain of one of his patients. These plaques are thought to be the drivers of the overall cognitive decline that is observed in AD. AD is now one of the leading causes of death in many developed countries, including the United Kingdom.

Optimal transport, trajectory inference, and lineage tracing - a case study by Aden Forrow

Towards the end of the eighteenth century, French mathematician and engineer Gaspard Monge considered a problem. If you have a lot of rubble, you would like to have a fort, and you do not like carrying rocks very far, how do you best rearrange your disorganised materials into organised walls? Over the two centuries since then, his work has been developed into the rich mathematical theory of optimal transport.

The skeleton in the phase diagram - a research case study by Nick Jones

In quantum many body physics, we look for universal features that allow us to classify complex quantum systems. This classification leads to phase diagrams of quantum systems. These are analogous to the familiar phase diagram of water at different temperatures and pressures, with ice and vapour constituting two phases. Quantum phase diagrams correspond to the different phases of matter at zero temperature, where the system is in its lowest energy state (usually called the ground state).

Quantum invariants via the topology of configuration spaces - Cristina Anghel

Knot theory studies embeddings of the circle into the three dimensional space and the first knot invariant was the Alexander polynomial. The world of quantum invariants started with the milestone discovery of the Jones polynomial and was expanded by Reshetikhin and Turaev’s algebraic construction which starts from a quantum group and leads to link invariants.

Freedom with boundaries

Oxford Mathematician Connor Behan discusses the ways in which a free quantum field can be coupled to a spatial boundary. His recent work with Lorenzo di Pietro, Edoardo Lauria and Balt van Rees sheds light on this question using the non-perturbative bootstrap technique.

Arbitrage-free neural-SDE market models

Oxford Mathematicians Samuel N. Cohen, Christoph Reisinger and Sheng Wang have developed new methods to help machine learning build economically reasonable models for options markets. By embedding no-arbitrage restrictions within a neural network, more trustworthy and realistic models can be built, allowing for better risk management in the banking system.

How compliance might help us get back to the workplace after lockdown

Oxford Mathematician Arkady Wey discusses a stochastic agent-based model of the workplace, developed to explore the importance of compliance with test and trace programs following a pandemic lockdown.

An agent-based simulation of the insurance industry: the problem of risk model homogeneity

Figure illustrating the structure of the model and the risk event scenario

In the 1680s there was a coffee house in London by the name of "Lloyd's". This place, catering to early maritime insurers, lent its name to the nascent English insurance market place, the now famous Lloyd's of London.

Machine learning with neural controlled differential equations

Oxford Mathematician Patrick Kidger writes about combining the mathematics of differential equations with the machine learning of neural networks to produce cutting-edge models for time series.

Can accounting for cross-immunity between related virus strains help us better forecast the spread of epidemics?

As we have seen over the past year, the ability to accurately predict the course of an epidemic is imperative when making decisions about how to control the spread of a virus.

Ambient and intrinsic geometry of Teichmüller spaces

Oxford Mathematician Vladimir Markovic talks about his research into intrinsic geometry of Teichmüller Spaces.

Tissue oxygenation and the growth of cancerous tumours

Tissue oxygenation plays a crucial role in the growth of cancerous tumours and their response to treatments. While it may seem intuitive that reducing oxygen delivery to a tumour would be a treatment therapy, low oxygen levels (hypoxia) can significantly reduce the effectiveness of treatments such as radiotherapy and some chemotherapies. Therefore, understanding the dynamics of a tumour's red blood cells - which carry oxygen through the vasculature - is of vital importance.

Re-examining geometry: p-adic numbers and perfectoid spaces

A tower of modular curves

Oxford Mathematician Daniel Gulotta talks about his work on $p$-adic geometry and the Langlands program.

"Geometry is one of the more visceral areas of mathematics. Concepts like distance and curvature are things that we can actually see and feel.

COVID-19 incidence is inversely proportional to T-cell production

One of the great puzzles of the current COVID-19 crisis is the observation that older people have a much higher risk of becoming seriously ill. While it is usually commonly accepted that the immune system fails progressively with age, the actual mechanism leading to this effect was not fully understood. In a recent work, Sam Palmer from Oxford Mathematics and his colleagues in Cambridge have proposed a simple and elegant solution to this puzzle.

Hawking Points in the Cosmic Microwave Background - a challenge to the concept of Inflation

For thirty years Oxford Mathematician Roger Penrose has challenged one of the key planks of Cosmology, namely the concept of Inflation, now over 40 years old, according to which our universe expanded at an enormous rate immediately after the Big Bang. Instead, fifteen years ago, Penrose proposed a counter-concept of Conformal Cyclic Cosmology by which Inflation is moved to before the Big Bang and which introduces the idea of preceding aeons.

How to design the perfect face mask – the effect of compressibility on filters

How do we design face masks that efficiently remove contaminants while ensuring that we can still breathe easily? One complicating factor with this question is the fact that the properties of the material that we start off with for our face mask can be very different when in use. A key example is seen when you stretch the mask around your face to put it on. In doing so, you also stretch the pores, i.e., the holes in the material that allow the air to pass through.

High dimensional footballs are almost flat

Oxford Mathematician Ben Green on how and why he has been pondering footballs in high dimensions.

"A 3-dimensional football is usually a truncated icosahedron. This solid has the virtue of being pleasingly round, hence its widespread use as a football. It is also symmetric in the sense that there is no way to tell two different vertices apart: more mathematically, there is a group of isometries of $\mathbf{R}^3$ acting transitively on the vertices.

The Oslo International Congress of Mathematicians in 1936 and the first Fields Medals

The International Congress of Mathematicians (ICM) that was held in Oslo in July 1936 was a unique event that took place in turbulent times, research by Oxford Mathematician Christopher Hollings has revealed.

Coronavirus Modelling - why social distancing works

Social distancing measures to reduce the spread of the novel coronavirus are in place worldwide. These guideline are for everyone. We are all expected to reduce our contact with others, and this will have some negative impacts in terms of mental health and loneliness, particularly for the elderly and other vulnerable groups. So why should we follow measures that seem so extreme? The answer is simple. Social distancing works. It reduces transmission of the virus effectively and lessens the impact on already stretched healthcare services.

Symplectic duality - a research case study from Andrew Dancer

One of the main themes of geometry in recent years has been the appearance of unexpected dualities between different geometric spaces arising from ideas in mathematical physics. One famous such example is mirror symmetry. Another kind of duality, which I am currently investigating with collaborators from Oxford and Imperial College, is symplectic duality.

Population network structure of genetic algorithms

Modelling changes in infectiousness in COVID-19

Oxford Mathematician William Hart and former Oxford Mathematician Dr Robin Thompson (now an Assistant Professor at the University of Warwick) discuss their latest joint COVID-19 research (carried out with fellow Oxford Mathematician Philip Maini), using mathematical models to infer changes in infectiousness during SAR

A network model of labor market dynamics

Colourings without monochromatic arithmetic progressions

Oxford Mathematician Ben Green on a tale of conjectures, mistaken assumptions and eventual solutions: a tale of mathematics.

Bacterial quorum sensing in fluid flow

By pooling resources between cells, colonies of bacteria can exhibit behaviours far beyond the capabilities of an individual bacterium. For example, bacterial populations can encase themselves in a self-generated polymer matrix that shelters cells in the core of the population from the external environment. Such communities are termed “bacterial biofilms”, and show increased tolerance to antimicrobial treatments such as antibiotics.

Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time

Take a piece of rope and knot it as you wish. When you are done, glue the two extremities together and you will obtain a physical realisation of what mathematicians also call a knot: a simple closed curve in 3-dimensional space. Now, put the knotted rope on a table and take a picture of it from above. It is now a planar projection of your knot. The mathematical equivalent of it is a knot diagram with multiple crossings as shown in the figure.

Rethinking defects in patterns

Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.

Product Replacement algorithm and property (T)

Lifting the Newfoundland Travel Ban - a Story of Mathematics, Coronavirus and the Law

When Oxford Mathematician Alain Goriely was approached by his collaborator Ellen Kuhl from Stanford University to work on a travel restriction issue in Newfoundland he started a Coronavirus journey that ended up in the Canadian Supreme Court.

Local Topology for Anomaly Detection in Data

Mathematical modelling of COVID-19 exit strategies

Mathematical models have been used throughout the COVID-19 pandemic to help plan public health measures. Attention is now turning to how interventions can be removed while continuing to restrict transmission. Predicting the effects of different possible COVID-19 exit strategies is an important current challenge requiring mathematical modelling, but many uncertainties remain.

Changing attitudes towards ancient arithmetic: reconciling mathematics with Egyptology

Oxford Mathematician Christopher Hollings and Oxford Egyptologist Richard Bruce Parkinson explain how our interpretation of Egyptian Mathematics has changed over the past two centuries and what that says about how historians of mathematics approach their subject.

Two Conjectures of Ringel

The varied world of Gaussian Fields

Oxford Mathematicians Dmitry Belyaev and Michael McAuley explain the ubiquitous role of Gaussian Fields in modelling spatial phenomena across science, and especially in cosmology. This case-study is based on work with Stephen Muirhead at Queen Mary University of London (QMUL). 

Mobile phone data and Covid-19 - missing an opportunity?

The Coronavirus disease pandemic (COVID-19) poses unprecedented challenges for governments and societies around the world. In addition to medical measures, non-pharmaceutical measures have proven to be critical for delaying and containing the spread of the virus. However, effective and rapid decision-making during all stages of the pandemic requires reliable and timely data not only about infections, but also about human behaviour, especially on mobility and physical co-presence of people. 

Why does the risk of cancer and infectious diseases increase with age?

Oxford Mathematician Sam Palmer tackles a crucial issue in our understanding of the risks of serious diseases such as cancer.

Generalised Lie algebras and their applications - Lukas Brantner

What takes a mathematician to the Arctic?

What takes a mathematician to the Arctic? In short, context. The ice of the Arctic Ocean has been a rich source of mathematical problems since the late 19$^{th}$ century, when Josef Stefan, aided by data from expeditions that went in search of the Northwest Passage, developed the classical Stefan problem. This describes the evolution of a moving boundary at which a material undergoes a phase change. In recent years, interest in the Arctic has only increased, due to the rapid changes occurring there due to climate change.

Rational Neural Networks

Deep learning has become an important topic across many domains of science due to its recent success in image recognition, speech recognition, and drug discovery. Deep learning techniques are based on neural networks, which contain a certain number of layers to perform several mathematical transformations on the input.

How growing nerves in the brain behave like light rays

During the early growth of the brain, an extraordinary process takes place where axons, neurons, and nerves extend, grow, and connect to form an intricate network that will be used for all brain activities and cognitive processes. A fundamental scientific question is to understand the laws that these growing cells follow to find their correct target.

Logarithmic Riemann-Hilbert Correspondences

Adaptive Cancer Therapy: tackling cancer drug resistance by capitalising on competition

How to deal with resistance? This is the headline question these days with regards to COVID vaccines. But it is an important question also in cancer therapy. Over the past century, oncology has come a long way, but all too often cancers still recur due to the emergence of drug-resistant tumour cells. How to tackle these cells is one of the key questions in cancer research. The main strategy so far has been the development of new drugs to which the resistant cells are still sensitive.

Numerically solving parametric PDEs with deep learning to break the curse of dimensionality

Oxford Mathematician Markus Dablander talks about his collaboration with Julius Berner and Philipp Grohs from the University of Vienna. Together they developed a new deep-learning-based method for the computationally efficient solution of high-dimensional parametric Kolmogorov PDEs.

Generalising Leighton's Graph Covering Theorem

Oxford Mathematician Daniel Woodhouse talks about the theorem that motivates much of his research.

Modelling Removal of Sulphur Dioxide from Flue Gas

Oxford Mathematician Kristian Kiradjiev talks about his DPhil research, supervised by Chris Breward and Ian Griffiths in collaboration with W. L. Gore and Associates, Inc., on modelling filtration devices for removal of sulphur dioxide from flue gas.

Frontiers of secrecy - the story of Eve, Alice and Bob

Oxford Mathematician Artur Ekert describes how his research in to using Quantum properties for cryptography led to some very strange results.

The Erdős primitive set conjecture

A set of integers greater than 1 is primitive if no number in the set divides another. Erdős proved in 1935 that the series of $1/(n \log n)$ for $n$ running over a primitive set A is universally bounded over all choices of A. In 1988 he conjectured that the universal bound is attained for the set of prime numbers. In this research case study, Oxford's Jared Duker Lichtman describes recent progress towards this problem:

Hard problems and security proofs in a Quantum World

In modern Cryptography, the security of every cryptosystem is required to be formally proven. Most of the time, such formal proof is by contradiction: it shows that there cannot exist an adversary that breaks a specific cryptosystem, because otherwise the adversary would be able to solve a hard mathematical problem, i.e. a problem that needs an unfeasible amount of time (dozens of years) to be concretely solved, even with huge computational resources.

Invariant theory for Maximum Likelihood Estimation

Oxford Mathematician Anna Seigal talks about her work on connecting invariant theory with maximum likelihood estimation.

Exercise of employee stock options with privileged information

Executive stock options (ESOs) are contracts awarded to employees of companies, which confer the right to reap the profit from buying the company stock (exercising the ESO) at or before a fixed maturity time $T$, for a fixed price specified in the contract (the strike price of the ESO). ESOs are used to augment the remuneration package of employees, the idea being to give them an incentive to boost the company's fortunes, and thus the stock price, making their ESO more valuable.

How to reduce damage when freezing cells

Oxford Mathematician Mohit Dalwadi talks about his work on the modelling of cryopreservation.

Quasiconvexity and its role in the Calculus of Variations

Oxford Mathematician Andre Guerra talks about quasiconvexity and its role in the Calculus of Variations:

Case Studies and Films Archive