Wednesday, 1 December 2021

Machine learning as a new tool for mathematicians

Today, two Oxford mathematicians András Juhász and Marc Lackenby published a paper in Nature in collaboration with authors from DeepMind and with Geordie Williamson at the University of Sydney. The paper describes a new crossover between the fields of mathematics and machine learning, where tools from machine learning have been used to discover new patterns in mathematics.

The project that Juhász and Lackenby focused on was in the area of knot theory. Knots are just closed curves in 3-dimensional space. They now have a very well-developed theory, created by many mathematicians over the past 120 years. This theory has several distinct sub-branches, including the use of non-Euclidean 3-dimensional geometry and the use of 4-dimensional invariants. In collaboration with DeepMind, Juhász and Lackenby were able to discover new and unexpected connections between the non-Euclidean geometry of knots and their 4-dimensional invariants. This new connection is of real interest to topologists, but what makes it particularly distinctive is its discovery using machine learning, which was able to detect non-linear relationships between knot invariants. Because of the huge number of different knot invariants, it would have been difficult to make this mathematical progress without machine learning.

This new use of machine learning in mathematics is undoubtedly going to be very widely applicable. In addition to the proofs of new theorems in knot theory by Juhász and Lackenby, it was used by Geordie Williamson in collaboration with DeepMind to discover new results in representation theory. It seems likely that it will become a new tool in many mathematicians' toolkits.

András Juhász said: “Pure mathematicians work by formulating conjectures and proving these, resulting in theorems. But where do the conjectures come from? We have demonstrated that, when guided by mathematical intuition, machine learning provides a powerful framework that can uncover interesting and provable conjectures in areas where a large amount of data is available, or where the objects are too large to study with classical methods".

Marc commented: “It has been fascinating to use machine learning to discover new and unexpected connections between different areas of mathematics. I believe that the work that we have done in Oxford and in Sydney in collaboration with DeepMind demonstrates that machine learning can be a genuinely useful tool in mathematical research”.

You can watch a video of Marc discussing the work below. For more information contact Dyrol Lumbard (

Monday, 15 November 2021

PROMYS Europe Connect 2021

PROMYS Europe Connect 2021 saw a group of enthusiastic and high-achieving young mathematicians gather (online) in July and August for a four-week intensive summer programme designed to give them the experience of thinking deeply about mathematics in a community of similarly mathematically excited students and staff. In other circumstances, PROMYS Europe is a six-week residential programme in Oxford, organised by a partnership of PROMYS (Boston), Wadham College and the Mathematical Institute at the University of Oxford, and the Clay Mathematics Institute. Circumstances being what they were, the 2021 programme, PROMYS Europe Connect, was tailored for an online format, in which we sought to capture as much as possible of the essence of the traditional event.

PROMYS Europe Connect 2021 was attended by 28 students, 2 of whom were returning students who first took part in PROMYS Europe in 2019, and 9 undergraduate counsellors. Participants represented Austria, Bulgaria, Czech Republic, Germany, Hungary, Norway, Romania, Serbia, Spain, Sweden, Switzerland, the UK and Ukraine. Places are offered to students on the basis of their academic potential as demonstrated in their application; we waive some or all of the fee (which is already heavily subsidised) for students who would otherwise be unable to participate. The programme is funded and resourced by the PROMYS Europe partnership, and by further financial support from alumni of the University of Oxford and Wadham College, and from the Heilbronn Institute for Mathematical Research.

This year, as usual, the core of the programme was a Number Theory course, taught by Glenn Stevens (Boston University, founding Director of PROMYS) and Henry Cohn (Microsoft Research, MIT). Students are encouraged to discover as much as possible for themselves, through their work on daily problem sets and through careful individual mentoring by their counsellor.

Alongside this was a Group Theory course with a similar philosophy, taught by Vicky Neale (Oxford), primarily aimed at the returning students, but also well attended by the first years. The returning students also worked on a research project on elliptic curves. Alongside their own mathematical seminars on category theory and analytic number theory, the counsellors organised social activities to help the students get to know each other and to build the sense of the community that is so important to the programme. These were complemented by guest lectures, exposing students to a range of current mathematical research. Plans are already under way for summer 2022, when we expect to return to the in-person format.

Here’s what participants said after PROMYS Europe Connect 2021:

Thank you for organizing all this, I feel like I almost cannot grasp the impact that this programme has had on my life in these three years of participation. [Returning student]

The online experience was actually much more engaging than I had expected. [Student]

PROMYS Europe Connect has left me with so many fantastic memories. I have truly felt like I am part of an amazing community of mathematicians who are all so passionate about what they do. [Counsellor]

Sunday, 14 November 2021

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. The paste softens, flows, and then bakes at about 350°C; the baked portion of the paste generates the strong electric arcs that heat up the furnace to 2000°C and power the entire production process. Heat is supplied by fans blowing hot air along the walls, and induced in the steel tube by the current supplied by the copper tubes. A schematic of the electrode and its components is shown in Figure 1 below. 

Throughout the years, demand has required electrode sizes and equipment to be changed, which has led to an imbalance in the melting process, making the cylinders sink down into the liquid paste, which results in an electrode with non-uniform material properties. In particular, unwanted electrical induction near the copper tubes that supply the current (item 6 in Figure 1) can lead to such issues, resulting in breakages, fires, and explosions. Among other things, Elkem ASA are interested in understanding how to avoid situations which may lead to such problems. This has led to a productive industrial collaboration between Aasgeir Valderhaug and Ben Sloman at Elkem ASA, and Oxford Mathematicians Ian Hewitt, Peter Howell, and myself, Alissa Kamilova, with the aim of understanding the softening process of the paste and, for example, how changes in cylinder addition may affect it.


                                                                                                 Figure 1

Using asymptotic techniques, we have derived the extensional flow time dependent mathematical model coupled with temperature represented in the top right schematic in Figure 1 to describe the paste softening process. Our modelling efforts were initially inspired by previous collaborations between Elkem ASA and Oxford Mathematician Andrew Fowler. We consider the effects of hot air heating, applied on the casing as a function of x, and the unwanted electrical induction, a source term represented by the green shaded region. The location where the paste touches the casing for the first time, λ(t), is a free boundary so it is an output of the model and in particular, when the paste does not reach the casing in this domain, we interpret it as a catastrophic failure of the process. Cylinder addition effects are incorporated into the boundary condition on x=0 at the top of the domain. In practice, 1 to 3 cylinders are added daily, although at times up to two days are skipped. Our mathematical model can replicate examples of these addition schedules and examine the differences in the paste softening process.

In Figure 2 we show how the area A, velocity u, and temperature θ change as paste is softened, where the black line shows the evolution of the free boundary λ. The time-dependence of the pressure exerted by the added cylinders has the most immediate and dramatic effect on the velocity u. In contrast, the profiles of A and θ indicate that the paste only begins to soften and deform significantly around x=0.3. There are relatively small but noticeable fluctuations in A and θ within each day and not only at the addition times. Furthermore, we found that the large additions every 5th day cause cold paste to be pushed downwards with a large velocity, which causes λ to decrease (move upwards), producing the initial change in this free boundary. Because this paste is colder, however, it does not spread as easily, which causes λ to increase again temporarily, leading to the local minimum observed in the black line in Figure 2. Such behaviours could be observed in reality when cylinder addition days are skipped and our simplified model is a first step in understanding them.

In addition to this work, we have exploited the small aspect ratio ϵ of the electrode as well as the strong temperature-dependent viscosity to derive the simplified steady model depicted in the bottom right schematic in Figure 1. We found that a new distinguished limit emerges when the viscosity variation across the domain is of order 1/ϵ2, resulting in a new hybrid model that contain elements of both lubrication theory and extensional flow. The model describes an approximate plug flow of the cold cylinders down the centre of the domain while hotter low-viscosity paste flows along the sides, and shows good agreement with costly CFD computations of a more complex version of this problem.

Collaborations between industrial partners and mathematicians are not only beneficial for the specific application relevant to the company but also allow us to look at interesting mathematical problems from a fresh perspective, leading to research and innovation that is beneficial for both parties.



                                                                                                                Figure 2

Friday, 12 November 2021

Congratulations Vitaliy!

For your Industrial Fellowship from the Royal Academy of Engineering.

Dr Kurlin’s fellowship will be an industry-scale application of the recently developed continuous classification of all periodic crystals up to rigid motion, the strongest equivalence relation preserving solid crystalline structures. This classification will continuously quantify similarities between periodic crystals that were impossible to compare in the past because of differences in their chemical composition or symmetry.

You can read more about it here

Wednesday, 10 November 2021

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.

The electric currents and voltages over the furnace are some of the few measurements that are taken during furnace operation, and from these readings furnace operators attempt to infer the internal state of the furnace. However it is not well understood what the changes in these electrical measurements mean physically, and how they relate to the efficiency of the silicon production.   



Figure 1: Illustration of a silicon furnace, adapted from TH Hannesson; The Si Process Drawings (left), and a portion of the electric circuit diagram for the furnace, showing the resistive components (right).

One way to understand the electrical measurements is to compare them to mathematical models of the furnace system. The furnace may be modelled as an electric circuit, with the electrodes and raw materials viewed as components in the circuit, with known electrical resistances. One particularly difficult aspect of the electrical system is the electric arcs which carry the current through hot, gas-filled craters within the bed of material in the furnace. In these arc regions the gas is heated to around 20,000K and ionised to a plasma, so that the dissociated electrons and ions can carry the electrical current. The electromagnetic forces generate a high-velocity flow of the plasma, and since the ionisation of the plasma increases with temperature, the size and shape of the electrically-conductive region is determined by the interacting heat generation, convection, and losses. Within the electric-circuit furnace models, the arc resistances are generally assumed to depend on the oscillating voltage over the arc, and are given by empirical ODE models. However, these empirical models were designed for other applications, and do not capture all the features of furnace arcs.


Figure 2: Axisymmetric simulations of a MHD model for an electric arc, showing the current density (streamlines) and the electrical conductivity (colour, units S/m) for a tall arc (left) and shorter arc (right).

We are studying a magnetohydrodynamic (MHD) model for electric arcs, which couples the Navier-Stokes equations for the fluid flow and Maxwell’s equations for the electromagnetic fields with an equation of conservation of energy in the arc fluid. Some simulations of this model are shown in Figure 2. A dimensional analysis of this MHD model suggests that, in the silicon furnace, the heat radiation is the dominant heat loss mechanism from the arc. We have proposed a simple model for the arc resistance, assuming that the resistive heating and the radiative heat loss are the only important heat transfer effects. Compared with the empirical models, our model captures the high-current behaviour of the arcs more accurately, predicting a quasi-steady, rising current-voltage characteristic in the high-current periods of the alternating current cycle. Since our model is physically derived, we may also estimate sensible ranges for the model parameters, reducing the number of parameters in the electric-circuit models which must be fitted to data.


Figure 3: Schematics for the possible asymptotic structures of an electric arc, with conduction effects dominating over convection (left), and both convection and conduction effects important in determining the radial extent of the arc (right).

While our simple arc model has several benefits over the currently-used empirical models, it predicts a slightly too rapid decrease in the electrical resistance as the arc heats up. We hypothesise that this is because we have neglected the convection of heat from the simplified model. A more-detailed asymptotic analysis of the MHD arc model is ongoing work, using the method of matched asymptotic expansions. While heat radiation remains the dominant heat-loss mechanism, the size of the arc region is determined by either the conduction or the convection effects at the boundary, depending on the magnitude of the applied current. Schematics of the possible arc structure when conduction (left) and convection (right) effects determine the radial extent of the arc are shown in Figure 3. Through this asymptotic analysis, we hope to gain deeper understanding of how the coupled processes within electric arcs interact, and develop improved arc models for use in the electric models for the silicon furnace.

This work has been funded through the EPSRC Industrially Focused Mathematical Modelling Centre for Doctoral Training (InFoMM) in collaboration with Elkem ASA. Ellen Luckins is a Postdoctoral Research Associate in the Oxford Centre for Industrial and Applied Mathematics (OCIAM).

Thursday, 4 November 2021

The 2021 Oxford Mathematics Admissions Test (in 10 minutes)

We can't interview all our undergraduate applicants in the time available, so to help us decide who to shortlist, we set the Oxford Mathematics Admissions Test (MAT) which all applicants for Maths, Computer Science, or joint honours courses must take.

Yesterday, 3 November, 5000 aspiring students from all around the world took the MAT for entrance to Oxford (and other universities). Here, courtesy of our admissions guru, James Munro, are the answers in 10 minutes. And remember, as they say before the football results on TV, if you don't want to know the answers, look away now.

You can find more MAT support via James' Online Maths Club. Everyone is welcome.

Tuesday, 2 November 2021

Pembroke Black Academic Futures Scholarship in Mathematics

As part of the University of Oxford’s Black Academic Futures Scholarships, the Mathematical Institute and Pembroke College are delighted to invite talented UK Black or Mixed-Black students to apply for one fully funded postgraduate scholarship in 2022-2023 on one of the courses below:

DPhil in Mathematics or Centre for Doctoral Training in Mathematics of Random Systems.

More information, including how to apply

Sunday, 31 October 2021

The First Term

So, the first term at university. And, more specifically, the first mathematical term at Oxford. What's in store? Well, our students' mathematical experience in their first term (and beyond) comprises two parts: lectures and tutorials. How do they work?

Lectures cover eight courses in the first term. These range from subjects such as Complex Numbers and Linear Algebra to an Introductory Calculus course. You can now watch an example, a Geometry lecture on Isometries, below. A full list of publicly available lectures can be found on our YouTube Channel.

Alongside lectures are tutorials where students, usually in pairs, meet with their tutor to go through the relevant course problem sheets. These tutorials provide the opportunity to spend time thinking and talking about the mathematics. You can watch an example - filmed in Trinity College in 2019 - below the lecture.

And of course there is the pleasure of meeting 200 other first term mathematicians like yourself and working with them on problems and sharing experience. Some of our first year students are sharing those experiences on our Twitter, Facebook and Instagram pages over the coming weeks.


Monday, 11 October 2021

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.

The mathematical tool they showcase is multi-parameter persistent homology (MPH) landscapes, developed by a recent Centre for Topological Data Analysis student, Oliver Vipond. MPH landscapes is a stable and computable invariant for studying data with multiple parameters using topological data analysis (see a previous article on MPH landscapes by Urike Tillmann). This is the first study to apply MPH landscapes to genuine real world applications.

Using a mechanistic model, the team generate synthetic spatio-temporal data of immune cells infiltrating a tumour spheroid, where the ground truth is known (Figure 1A, top). They then apply MPH landscapes to the data and show how it surpasses single parameter persistent homology (Figure 1B). This synthetic study also validates the interpretation of MPH landscapes (Figure 1C).



Next, the team showcase MPH landscapes on slices of cancer tissue (Figure 1A, bottom). They show how MPH landscapes can quantify, characterise, and compare the distributions of different immune cell subtypes as they are infiltrating into tumours, and how these patterns change in different environmental conditions such as hypoxia (i.e., low oxygen levels). The team propose that one of the parameters, i.e., co-density in the bi-filtration (Figure 1B, bottom), is a good proxy for the oxygen level within the tissue. As the point clouds are too large for a single computation, they take multiple subsamples. Statistical analysis of the MPH landscapes suggests that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages (Figure 1C, bottom). This study highlights how Topological Data Analysis can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation.

This work is truly interdisciplinary: the mechanistic models generate synthetic data that is used as a ground truth to validate and interpret the utility of the methods, which gives greater confidence to apply them to study the immune/tumour microenvironment. This work highlights the power of MPH landscapes for quantifying, characterising, and comparing features within the tumour microenvironment in synthetic and real datasets and the benefits of working across disciplines.

You can read more about this work which appears in Proceedings of the National Academy of Sciences (PNAS), co-authored by Oxford mathematicians, clinicians and pathologists: Oliver Vipond, Joshua Bull, Philip Macklin, Ulrike Tillmann, Christopher Pugh, Helen Byrne and Heather Harrington (pictured top).

Sunday, 10 October 2021

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.

The (A$\beta$) and ($\tau$P) proteins occur naturally in the brain but not in an aggregated state. It is now thought that a prion-like mechanism, a process similar to prion diseases [4], lies at the heart of this pathology [1,2]. The prion process can be summarized in two steps. First, a misfolded seed protein joins with a nearby healthy one. The misfolded protein 'transmits' its toxic misfolded state to the healthy protein. The pair then break apart into two separate toxic, misfolded seeds. The prion mechanism can be modelled with a few simple equations.

A few more salient things are at play in the brain that we will want to include in our simplified model. For one, healthy proteins are constantly being produced, and removed, in the brain. In addition, misfolded proteins do occur naturally but the brain has several means of removing them. If we ignore the complexities of brain facilities management, we can write down a simple model of organ-wide proteostasis that reflects these basic mechanisms of AD.



                                                               Figure 1: A composite brain graph constructed from the data of 492 patients

Suppose that the brain is a domain $\Omega$ and that $u$ and $\tilde{u}$ are healthy and toxic (misfolded) protein concentrations, respectively. The basic AD model is then \begin{align}\label{eqn:hxdimer-base} \dot{u} &= a_0 - a_1 u - a_{2}u\tilde{u},\\ \dot{\tilde{u}} &= -\tilde{a}_1 \tilde{u} + a_{2}u\tilde{u}, \end{align} where $a_0$ is the healthy production rate, $a_1$ and $\tilde{a}_1$ are rates of healthy and toxic protein removal and $a_{2}$ is the rate that toxic proteins recruit healthy proteins into the toxic state. As $a_0$, $a_1$, $a_2$ and $\tilde{a}_1$ correspond to physical systems, they take on positive values. The dynamics of this system are simple but biologically insightful. First, there are two stationary states \[ \left(u_1, \tilde{u}_1\right) = \left(\frac{a_0}{a_1}, 0\right)\quad\text{and}\quad  \left(u_2, \tilde{u}_2\right) = \left(\frac{\tilde{a}_1}{a_{2}}, \frac{a_0a_{2} - a_1\tilde{a}_1}{\tilde{a}_1 a_{2}}\right). \]

Biologically, the first state, $(u_1,\tilde{u}_1)$, describes the presence of healthy protein and the absence of the toxic variant. This is the healthy brain that we enjoy the benefits of for most of our lives. The second stationary state represents the presence of a toxic protein infection and, in our current context, is the AD brain. What does our model say about our brain's health? The eigenvalues of the Jacobian, of the system above, for the healthy brain are \[ \lambda_1 = -a_1\quad\text{and}\quad \lambda_2 = \frac{a_{2}a_0 - \tilde{a}_1 a_1}{a_1}. \]

These eigenvalues mean that our healthy brain stays healthy only if\begin{equation}\label{eqn:clearance-ineq} a_{2}a_0 < \tilde{a}_1 a_1. \end{equation} The inequality above says, loosely, that when protein production rates are controlled by the clearance rates then the brain is safe from AD. Our model suggests that the clearance systems of the brain may play a special role in AD. In fact, it turns out that this is a non-trivial, but well-studied, clinical observation [3,5]. This is a good indication that our models are on the right track and can be extended to include more complex effects.



       Figure 2; AD progression for a simulated patient. Toxic (A$\beta$) protein (left) and ($\tau$P) protein (right)

Until this point, we have looked at the brain as a single domain. However, we know from postmortem studies that the toxic protein infection starts in one brain region and then spreads. In fact, clinical experiments suggest [2] that proteins move around the brain using the fibrous bundles of the brain's axonal connections. Let us consider the individual cortical and subcortical regions of the brain as separate domains, $\Omega_i$. We can think of the brain as a very large graph where the vertices are the $\Omega_i$ and the edges represent the axons connecting the regions. Figure 1 shows a graph, with 1,015 vertices and 70,892 edges, constructed from the neuroimaging data of 492 patients. The final piece of the puzzle comes from the observation that the axonal connections are the dominant means of protein traffic, from region to region, in the brain. Our initial, static, model can be augmented with transport dynamics. Let $u_i$ and $\tilde{u}_i$ denote the healthy and toxic concentrations in $\Omega_i$. A Laplacian matrix, $\mathbf{L} = L_{ij}$, introduces transport along the axons as \begin{align}\label{eqn:hxdimer-base:diffusion} \dot{u}_i &= -\rho_u \sum_{j=1}^{N} L_{ij} u_{ij} + a_0 - a_1 u_i - a_{2}u_i\tilde{u}_i,\\ \dot{\tilde{u}}_i &= -\rho_{\tilde{u}}\sum_{j=1}^{N} L_{ij}\tilde{u}_{ij} -\tilde{a}_1 \tilde{u}_i + a_{2}u_i\tilde{u}_i. \end{align}

Models like the one above are the basic starting point for further studying AD dynamics. Extensions of this simple model have already been used to test whether AD mechanisms, observed in mice, might also play a role in humans. For instance, interactions between (A$\beta$) and ($\tau$P) in AD are thought to be critical for understanding the death of neurons and the development of atrophy. Modeling suggests [6] that these interactions may indeed be central to disease progression in humans and that interacting models can explain patient neuroimaging data.

Our simple model suggested to us that the brain's clearance systems played a specific role in protecting us from AD. In fact, more elaborate modeling can show [7] how the brain's clearance systems fight back against the toxic seed proteins as they try to aggregate to form the plaques that Alzheimer first observed. Mathematical models of AD also afford us the ability to use statistical techniques to obtain personalized, predictive simulations from patient data. Figure 2 shows the toxic (A$\beta$) and ($\tau$P) distribution for a simulated AD patient.

Oxford Mathematics is taking a stand against AD by using mechanistic models and scientific computing to bring a personalized, predictive clinical understanding of the disease to the UK. That's something you can't forget.

Travis Thompson is a Postdoctoral Research Associate at the Mathematical Institute in Oxford.

[1] Michel Goedert. Alzheimer’s and Parkinson’s diseases: The prion concept in relation to assembled aβ, tau, and α-synuclein. Science, 349(6248):1255555, 2015.

[2] G. Hallinan, M. Vargas-Caballero, J. West, and K. Deinhardt. Tau misfolding efficiently propagates between individual intact hippocampal neurons. J. Neurosci., 2019.

[3] M. Nedergaard and S. Goldman. Glymphatic failure as a final common pathway to dementia. Science, 370(6512):50–56, 2020.

[4] Stanley B Prusiner. Prions. Proceedings of the National Academy of Sciences, 95(23):13363– 13383, 1998.

[5] Jenna Tarasoff-Conway, Roxana Carare, and Mony J. de Leon et al. Clearance systems in the brain–implications for Alzheimer disease. Nat Rev Neurol, 11(8):457–470, 2015.

[6] T.B. Thompson, P. Chaggar, E. Kuhl, and A. Goriely. Protein-protein interactions in neurodegenerative diseases: a conspiracy theory. PLoS Comp. Biol., 16(10):e1008267, 2020.

[7] T.B. Thompson, G. Meisl, and A. Goriely. The role of clearance mechanisms in the kinetics of toxic protein aggregates involved in neurodegenerative diseases. J. Chem. Phys., 154:125101, 2021.