Congratulations to Oxford Mathematicians Vicky Neale and Ursula Martin who have been nominated for Suffrage Science awards. The awards celebrate women in science and encourage others to enter science and reach senior leadership roles. The 11 awardees are chosen by the previous award holders and the awards themselves are items of jewellery, inspired by the Suffrage movement, and are passed on as heirlooms from one female scientist to the next.

Ursula was nominated by Professor Dame Wendy Hall, University of Southampton and Vicky was nominated by Professor Dame Celia Hoyles, University College London.

Oxford Mathematician Siddharth Arora talks about his and his colleagues' research in to using smartphone technology to anticipate the symptoms of Parkinson’s disease.

"Parkinson’s disease (PD) is the second most common neurodegenerative disease, the hallmarks of which include tremor, stiffness, and slowness of movement. Existing tests for the assessment of Parkinson’s require in-clinic examination of symptoms by a clinician. This can sometimes cause a delay in diagnosis. It is believed that there are changes in the brain 5 to 10 years before the symptoms of PD become evident.

To try and facilitate early diagnosis of Parkinson’s, together with the Oxford Parkinson's Disease Centre (OPDC) we investigated if smartphones can be used to detect any potential differences in motor symptoms associated with PD and REM Sleep Behaviour Disorder (RBD). It is now increasingly recognised that having RBD may be a risk factor for developing future Parkinson’s. In this study, we used a smartphone app featuring 7 motor tests to measure: (1) Voice, (2) Balance, (3) Gait, (4) Finger tapping, (5) Reaction time, (6) Rest tremor, and (7) Postural tremor. Recordings were collected both in-clinic and at home. Using the smartphone recordings, we extracted key features of interest and used machine learning to quantify patterns of motor impairment that are specific to RBD, PD, and Controls. In one of the largest cohorts of deeply phenotyped participants, we report that smartphones can be used to discriminate between participant groups with a high level of accuracy (84.6% to 91.9% mean sensitivity and specificity). Our research paper focussing on ‘detecting the early motor symptoms’ of Parkinson’s is published here. A pilot study on ‘monitoring the severity of PD symptoms’ can also be accessed here, while a large-scale study using the mPower data to understand the ‘longitudinal characteristics’ of Parkinson's through the analysis of finger tapping and memory tests is published here.

Moreover, we also investigated potential vocal deficits in people who are at an increased risk of Parkinson’s (carriers of LRRK2 mutation). Our preliminary findings suggest that vocal deficits in LRRK2-associated PD may be different than those in idiopathic Parkinson’s. This research could help develop an inexpensive remote screening test for assessing the risk LRRK2-associated Parkinson’s based on voice - click here for the article.

These findings provide an exciting and growing consensus for the utility of digital biomarkers in early and pre-symptomatic Parkinson’s."

Free suspended liquid films or sheets are often formed during industrial production of sprays as well as in natural processes such as sea spray. Early experimental and theoretical investigations of them were done by French physicist Felix Savart, who observed liquid sheets forming by a jet impact on a solid surface, or by two jets impacting each other (1833), and British physicist Arthur Mason Worthington, a pioneer in investigation of the crown splash forming after impact of a drop onto a liquid surface. Worthington observed free liquid fims forming in splashes in the form of ejecta sheets (1908).

The industrial production of sprays proceeds typically via the formation of sheets, which break up at the edges to form ribbons. Ribbons are susceptible to the Rayleigh-Plateau instability, and quickly break up into drops. Liquid sheets also happen to puncture far from their boundaries, nucleating a hole or a collection of adjacent holes on the plane of the film. An intriguing example of such hole formation in a crown splash was observed experimentally. Here the whole liquid sheet forming initially the crown breaks up into soap network of filaments.

It is therefore of crucial importance to understand the mechanisms leading to the breakup of sheets. In contrast to jets and liquid threads there is no obvious linear mechanism for sheet breakup. Average thickness of a sheet at breakup start has been measured to lie between 100nm and 100μm, depending on the purity of the liquid, and therefore, van der Waals force cannot play a significant role for breakup formation except perhaps in its very last stages.

Figure 1: This image is a watercolor painting by illustrator Anca Pora following the experimental work of Thoroddsen et al (J. Fluid Mech., 557, 63-72) and provided by Dr. Radu Cimpeanu (Oxford Mathematics).

Figure 2: Typical profiles of the height (black), velocity (blue) and temperature (red) for a free liquid sheet

Figure 3: Height profiles in the thinning patch (left) and the sharp forming jump in the temperature (right).

Oxford Mathematician Georgy Kitavtsev together with his collaborators at the University of Bristol and ICMAT, Madrid confirmed recently that variations of temperature or alternatively impurities at the sheet free surfaces (such as diffusing surfactants) promote breakup, because they produce Marangoni forces, which lead to non-steady flows. Intriguingly, even slight variations of temperature or surfactant distribution can lead to formation of sharp gradients of them that promote consequently sheet rupture in an infinite time at an exponential fast rate (see Figs. 3-4). Using matched asymptotic analysis of the underlying nonlinear system of coupled PDEs the researchers were able to derive analytically the corresponding structure of the liquid sheet undergoing rupture. Consistent with the above experimental observations the found solution demonstrates formation of an exponentially thinning patch inside of the sheet and accumulation of the fluid mass into the forming filaments.

In two subsequent papers the researchers looked now at evolution of pure viscous sheets and showed that generic solutions to a PDE system describing their evolution rather exponentially asymptotically decay to the flat profile. By that they provided a proof of the conjecture formally accepted in the physical literature that a viscous sheet cannot rupture in a finite time in the absence of external forcing. In this proof, a transformation of the PDE system into Lagrangian coordinates turned out to be very useful. Furthermore, in the absence of or for negligible surface tension the Lagrange system is given by a single singular diffusion equation with a source term determined solely by the initial height and velocity distribution in the sheet. In this case, an exponential asymptotic convergence of solutions to the latter equation to the non-homogeneous limit profiles was observed firstly numerically (see Fig. 5) and understood analytically.

Figure 4: Log-linear plot of the minimum height of the sheet as function of time.

Figure 5: Convergence to non-homogeneous stationary height profiles of free viscous liquid sheets in the absence of surface tension.

In summary, these mathematical results open several new perspectives for future industrial applications of free thin liquid sheets and understanding of
their complicated behaviour.

Welcome to our new undergraduate students, young mathematicians of diverse nationalities from Afghan to Kazakh, Syrian to Pakistani, Malaysian to Greek. And 70% of UK students from state schools. Welcome to Oxford Mathematics, all of you.

And watch out for a snippet or two from their lectures next week on our Twitter and Facebook pages. We hope it will inspire those of you who hope to join them in the future.

Over the last few years, the study of the physiological mechanisms governing the movement of fluids in the brain (referred to as the brain waterscape) has gained prominence. The reason? Anomalies in the brain fluid dynamics are related to diseases such as Alzheimer's disease, other forms of dementia and hydrocephalus. Understanding how the brain waterscape works can help discover how these diseases develop. Unfortunately, experimenting with the human brain in vivo is extremely difficult and the subject is still poorly understood. As a result, this topic is a highly active interdisciplinary research area. In our research team, formed by Oxford Mathematicians Matteo Croci,Patrick Farrell and Mike Giles in collaboration with Marie E. Rognes, Vegard Vinje and colleagues from Simula Research Laboratory, we contribute by simulating the brain waterscape on a computer, hence providing an alternative avenue of investigation that is cheap and does not require human experimentation.

One of the main challenges in brain simulation is the lack of accurate quantitative information on the mechanical input parameters needed to set up our mathematical model. Quantities such as brain matter permeability, interstitial fluid flow velocity and diffusivity are only known approximately or on average. The position of the blood vessels and capillaries can be measured, but it varies from patient to patient and it is extremely difficult to resolve without significant expense.

To overcome these issues, we can construct surrogates for these quantities that account for the uncertainty in their values through the use of Gaussian random fields. Gaussian random fields are functions of space whose values at each point are given by Gaussian random variables which are correlated according to a given covariance function. Sampling realisations of these fields can be extremely expensive computationally. Our team of researchers developed one of the fastest currently available sampling algorithms for Mat\'ern-Gaussian fields. This is achieved by recasting the Gaussian field sampling problem as the solution of an elliptic partial differential equation (PDE) driven by spatial white noise. Solving this equation is standard, but sampling white noise is not. The developed algorithm is able to draw white noise realisations quickly and efficiently and can be used in conjunction with the multilevel Monte Carlo (MLMC) method for further acceleration (see here). The coupling of white noise required by MLMC is enforced even in the non-nested case thanks to a supermesh construction. Furthermore, the algorithm can be employed to solve a wider class of PDE problems with spatial white noise additive terms.

For further information on the efficient sampling and coupling of white noise for MLMC with non-nested meshes see here. A paper on uncertainty quantification in brain simulation is in preparation and a link will appear here in due course.

Amid all the debate about equipping ourselves for the 'technological' world of the future, one thing is for sure: the quality of research in the Mathematical and Life Sciences (and beyond) depends on the quality of its young researchers. In that spirit we are delighted to welcome our latest cohort of DPhil (PhD) students, 43 of them, all fully funded, from across the globe. 13 from the UK, 15 from the European Union and a further 15 from India, Kenya, Norway, Australia, Mexico, USA, China, Switzerland, Argentina, Israel and South Africa.

We would also like to welcome our masters students and in particular our first cohort of Oxford Masters in Mathematical Sciences (OMMS) students, 36 in total, 26 men and 10 women from 17 different countries. This standalone course offers students the opportunity to join our current fourth year undergraduates and work with our internationally renowned faculty.

Welcome to everyone. We (and we don't just mean Oxford) need you all.

Roger Penrose's relationship with the artist M.C. Escher was not just one of mutual admiration. Roger's thinking was consistently influenced by Escher, from the famous Penrose tiling to his groundbreaking work in cosmology. The respect was mutual, as was clear when Roger dropped in to see Escher at his home...

Oxford Mathematics hosted this special event in its Public Lecture series during the conference to celebrate the 20th Anniversary of the foundation of the Clay Mathematics Institute.

Oxford Mathematician Kristian Kiradjiev has won the Graham Hoare Prize (awarded by the Institute of Mathematics and its Applications) for his article "Connecting the Dots with Pick's Theorem". The Graham Hoare Prize is awarded annually to Early Career Mathematicians for a brilliant Mathematics Today article. Kristian also won the award in 2017. Here he talks about his work.

"Pick's theorem is an example of a theorem that is not widely known but has surprising applications to various mathematical problems. At its essence, Pick's theorem is a geometrical result, but has quite a few algebraic implications.

Published in 1899 by Georg Alexander Pick, the theorem states that given any simple polygon whose vertices lie on an integer grid, its area, $A$, is calculated according to the following formula: \begin{equation} A=i+\frac{b}{2}-1, \label{eq:Pick} \end{equation} where $i$ is the number of grid points inside the polygon, $b$ is the number of grid points that lie on the boundary of the polygon, and by simple we mean a polygon without holes. For a polygon with $n$ holes, the formula generalises to \begin{equation} A=i+\frac{b}{2}-1+n. \label{eq:Pick1} \end{equation} A useful property of Pick's formula that we immediately note is that it is invariant under shearing of the lattice. Also, scaling the distance between grid points in one direction simply scales the area of the polygon. These two observations can be used to show that Pick's formula is valid for sheared (triangular) grids as well, provided we take account of any scaling in the direction perpendicular to the shearing one.

Pick's theorem can be used to tackle a number of problems in different fields of mathematics. To give a flavour of this, we present a geometric problem that is simple to state. Although square integer grids are ubiquitous in our life, it is a fact that one cannot draw one of the simplest figures, namely, an equilateral triangle with its vertices being grid points on such a lattice. Some standard proofs involve tedious algebra and trigonometry, whereas Pick's theorem proves the result immediately. Suppose we have drawn an equilateral triangle with sides of length $d$ on a square grid. Then, its area is given by the well-known formula $A=\sqrt{3} d^2/4$. Since the vertices of the triangle are integer points, then $d^2$ is an integer (by Pythagoras' theorem, for example), and, thus, the area is an irrational number. However, Pick's formula on square grids always gives a rational area. This contradiction proves the initial claim. The same argument shows that regular hexagons cannot be drawn on a square integer grid either. Another problem where Pick's theorem can be applied concerns some properties of the so-called Farey sequences.

One might think that Pick's theorem can be easily generalised to three (or more) dimensions for volume of solids, etc. However, in 1957, J. Reeve produced an example of what is now known as the Reeve tetrahedron, which shows that Pick's theorem does not have a direct analogue in three or more dimensions by devising a figure that can take many different values for its volume without changing the number of interior and boundary points. However, Pick's theorem has some close analogues in more than two dimensions such as the Ehrhart polynomials.

The figure above is a rather complicated polygon, called a Farey sunburst (because of its relation to Farey sequences). Its area can be readily calculated using Pick's theorem to be 48 square units."

Helen will be the recipient of the Leah Edelstein-Keshet Prize for her work focused on the development and analysis of mathematical and computational models that describe biomedical systems, with particular application to the growth and treatment of solid tumors, wound healing and tissue engineering. This award recognizes an established scientist with a demonstrated track record of exceptional scientific contributions to mathematical biology and/or has effectively developed mathematical models impacting biology. "Dr. Byrne has made outstanding scientific achievements coupled with her record of active leadership in mentoring scientific careers." The Edelstein-Keshet Prize consists of a cash prize of $500 and a certificate given to the recipient. The winner is expected to give a talk at the Annual Meeting of the Society for Mathematical Biology in Montreal in 2019.

Francis has won the H. D. Landahl Mathematical Biophysics Award. This award recognizes the scientific contributions made by a postdoctoral fellow who is making exceptional scientific contributions to mathematical biology. The award is acknowledged with a certificate, and a cash prize of USD $500.

Oxford Mathematics and the Clay Mathematics Institute Public Lectures

Roger Penrose - Eschermatics

Roger Penrose’s work has ranged across many aspects of mathematics and its applications from his influential work on gravitational collapse to his work on quantum gravity. However, Roger has long had an interest in and influence on the visual arts and their connections to mathematics, most notably in his collaboration with Dutch graphic artist M.C. Escher. In this lecture he will use Escher’s work to illustrate and explain important mathematical ideas.