Monday, 26 November 2018

Xenia de la Ossa awarded the Dean’s Distinguished Visiting Professorship by the Fields Institute in Toronto

Oxford Mathematician Xenia de la Ossa has been awarded the Dean’s Distinguished Visiting Professorship by the Fields Institute in Toronto and the Mathematics Department of Toronto University for the Fall of 2019.  Xenia will be associated with the thematic programme on Homological algebra of mirror symmetry.

Xenia's research interests are in Mathematical Physics, Geometry and Theoretical Physics, specifically in the mathematical structures arising in String Theory. 

Friday, 23 November 2018

The fluid mechanics of kidney stone removal

The discomfort experienced when a kidney stone passes through the ureter is often compared to the pain of childbirth. Severe pain can indicate that the stone is too large to naturally dislodge, and surgical intervention may be required. A ureteroscope is inserted into the ureter (passing first through the urethra and the bladder) in a procedure called ureteroscopy. Via a miniscule light and a camera on the scope tip, the patient’s ureter and kidney are viewed by a urologist. The field-of-view is obstructed by blood and stone particles, so a saline solution flows from a bag hanging above the patient, through a long, thin channel (the working channel) that runs through the shaft of the ureteroscope. The fluid flows out of the scope tip, clearing the area in front of the camera and exits the body, flowing in the opposite direction along the outside of the scope through an access sheath, a rigid tube that surrounds the scope. Stones are removed by auxiliary working tools of varying sizes which are introduced through the working channel, providing an undesirable resistance to the flow.

The flow of saline solution is vital for successful ureteroscopy, and understanding and improving this process, known as irrigation, is the subject of this research, carried out by a team comprising Boston Scientific, a medical manufacturing company, Ben Turney, a urologist based at the Nuffield Department of Surgical Sciences in Oxford, and Oxford Mathematicians Sarah Waters, Derek Moulton, and Jessica Williams.

The team apply mathematical modelling techniques to ureteroscope irrigation, based on systematic reductions of the Navier-Stokes equations. Due to the resistance to flow created by working tools, there is a complex relationship between driving pressure, scope geometry, and flow rate properties. The objective has been to understand and exploit that relationship to increase flow for a given driving pressure drop. The team have shown that increased flow and decreased kidney pressures can be accomplished through the use of non-circular cross-sectional shapes for the working channel and the access sheath. These results have led to the filing of a joint patent with Boston Scientific. To complement the reduced analytical models, the team are performing numerical simulations to gain further insight into the flow patterns and resulting pressures within the kidney for a given operating set-up.

Due to the real-world application of this modelling, it is vital that the predictions are validated via experiments. The researchers have performed bench-top flow tests to confirm their analytical models, and particle imaging velocimetry (PIV) to compare against their numerical simulations for the flow within the kidney. This work in constructing a mathematical framework to describe ureteroscope irrigation has significant potential in quantifying irrigation flow and improving scope design.



Left: A diagram of the urinary system. The ureteroscope is inserted into the urethra, passing through the bladder, ureter, and into the kidney.

Right: An idealised ureteroscopy set-up. The bag of saline solution is at a height, above the patient. The ureteroscope shaft, containing a working channel, is inserted into the patient. The fluid is driven through the working channel with flow rate by the applied pressure drop, and returns back through an access sheath.


Left: The predicted flow rate through a working channel of circular cross-section containing a working tool of circular cross-section (shaded region). Upper black line of the working tool is at the edge of the channel, the lower line is in the centre. This is compared with experimental data from bench-top experiments (red data points). The dashed and dotted lines are for working channels of elliptical cross-sections with elliptical eccentricity values 0.53 and 0.71. The working tool is in the position that optimises the flow (at the edge of the channel).

Right: Streamlines for simulated flow exiting the working channel into the kidney and returning back through the access sheath. Computed using open-source finite element library oomph-lib.

Thursday, 8 November 2018

Simulating polarized light

The Sun has been emitting light and illuminating the Earth for more than four billion years. By analyzing the properties of solar light we can infer a wealth of information about what happens on the Sun. A particularly fascinating (and often overlooked) property of light is its polarization state, which characterizes the orientation of the oscillation in a transverse wave. By measuring light polarization, we can gather precious information about the physical conditions of the solar atmosphere and the magnetic fields present therein. To infer this information, it is important to confront observations with numerical simulations. In this brief article we will focus on the latter.

The transfer of partially polarized light is described by the following linear system of first-order coupled inhomogeneous ordinary differential equations (ODEs) \begin{equation} \frac{\rm d}{{\rm d} s}\mathbf I(s) = -\mathbf K(s)\mathbf I(s) + \boldsymbol{\epsilon}(s)\,. \label{eq:RTE} \end{equation}

In this equation, the symbol $s$ is the spatial coordinate measured along the ray under consideration, $\mathbf{I}$ is the Stokes vector, $\mathbf{K}$ is the propagation matrix, and $\boldsymbol{\epsilon}$ is the emission vector.

The analytic solution of this system of ODEs is known only for very simple atmospheric models, and in practice it is necessary to solve the above equation by means of numerical methods.

Although the system of ODEs in the equation above is linear, which simplifies the analysis, the propagation matrix $\mathbf{K}$ depends on the spatial coordinate $s$, which implies that this system is nonautonomous. Additionally, it exhibits stiffness, which means that extra care must be taken into account to compute a numerical solution because numerical instabilities are just around the corner, and these have the potential to completely invalidate numerical computations.

In their work Oxford Mathematician Alberto Paganini and Gioele Janett from the solar research institute IRSOL in Locarno, Switzerland, have developed a new algorithm to solve the equation above. This algorithm is based on a switching mechanism that is capable of noticing when stiffness kicks in. This allows combining stable methods, which are computationally expensive and are used in the presence of stiffness, with explicit methods, which are computationally inexpensive but of no-use when stiffness arises.

The following plots display the evolution of the Stokes components along the vertical direction for the Fe I line at 6301.50 Å in the proximity of the line core frequency (the Stokes profiles have been computed considering a one-dimensional semi-empirical model of the solar atmosphere, discretized on a sequence of increasingly refined grids). The black line depicts the reference solution, while the dots denote the numerical solution obtained with the new algorithm. Different dot colors correspond to different methods: Blue dots indicate the use of an explicit method, whereas yellow, orange, and purple dots indicate the use of three variants of stable methods (each triggered by a different degree of instability). These pictures (below) show that the algorithm is capable of switching and choosing the appropriate method whenever necessary and of delivering good approximations of the equation above.



This research has been published in The Astrophysical Journal, Vol 857, Number 2, p. 91 (2018).

Tuesday, 6 November 2018

Conformal Cyclic Cosmology. Roger Penrose and Hannah Fry - Oxford Mathematics London Public Lecture now online

He calls it a "crazy idea." Then again, he points out, so is the idea of inflation as a way of explaining the beginnings of our Universe.

In our Oxford Mathematics London Public Lecture at the Science Museum in London, Roger Penrose revealed his latest research. In both his talk and his subsequent conversation with fellow mathematician and broadcaster Hannah Fry, Roger speculated on a veritable chain reaction of universes, which he says has been backed by evidence of events that took place before the Big Bang. With Conformal Cyclic Cosmology he argues that, instead of a single Big Bang, the universe cycles from one aeon to the next. Each universe leaves subtle imprints on the next when it pops into being.  Energy can 'burst through' from one universe to the next, at what he calls ‘Hawking points.’

In addition to his latest research Roger also reflects on his own approach to his subject ("big-headedness") and his own time at school where he was actually dropped down a maths class. So we are not alone, universally or personally speaking.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Photos courtesy of the Science Museum Group.



Tuesday, 6 November 2018

Improving techniques for optimising noisy functions

The problem of optimisation – that is, finding the maximum or minimum of an ‘objective’ function – is one of the most important problems in computational mathematics. Optimisation problems are ubiquitous: traders might optimise their portfolio to maximise (expected) revenue, engineers may optimise the design of a product to maximise efficiency, data scientists minimise the prediction error of machine learning models, and scientists may want to estimate parameters using experimental data. In real-world settings, uncertainties and errors are unavoidable, and this can cause stochastic noise to be present in the objective.

Most methods for optimisation rely on being able to evaluate both the objective and its derivatives.  Access to first derivatives is important for finding uphill or downhill directions, which tell us where to search next for optima, and when to terminate the method. However, when the objective has stochastic noise, it is no longer differentiable, and standard optimisation methods do not work. Instead, we must develop ‘derivative-free’ optimisation methods; that is, we have to answer the question “how do you get to the top of a hill when you don’t know which way is up?”. We achieve this by constructing models of the landscape based on sampling objective values – this approach is based on rigorous mathematical principles, and has provable guarantees of success. The figure above shows a noisy landscape, and the points tested by a derivative-free method searching for the true minimum (bottom centre, in green).

Oxford Mathematicians Lindon Roberts and Coralia Cartis, together with Jan Fiala and Benjamin Marteau from Numerical Algorithms Group Ltd (NAG), a British scientific computing company, have developed a new derivative-free method for optimising noisy and expensive objectives. The method automatically detects when the information in the objective value is overwhelmed by noise, and kick-starts the method to bring more information into the models of the landscape. This approach requires fewer evaluations of the (possibly expensive) objective, runs faster and is more scalable, but produces as good solutions as other state-of-the-art methods. Their ideas are being commercialised by NAG and will soon be available in their widely-used software library. This technique is also being applied to parameter estimation for noisy climate simulations, to help scientists find optimal parameters that fit observational climate data, thus helping quantify the sensitivity of our climate to CO2 emissions.

This work is supported by the EPSRC Centre for Doctoral Training in Industrially-Focused Mathematical Modelling. 

Monday, 5 November 2018

Structure or randomness in metric diophantine approximation?

Diophantine approximation is about how well real numbers can be approximated by rationals. Say I give you a real number $\alpha$, and I ask you to approximate it by a rational number $a/q$, where $q$ is not too large. A naive strategy would be to first choose $q$ arbitrarily, and to then choose the nearest integer $a$ to $q \alpha$. This would give $| \alpha - a/q| \le 1/(2q)$, and $\pi \approx 3.14$. Dirichlet, introducing the pigeonhole principle, showed non-constructively that there are infinitely many solutions to $| \alpha - a/q| \le 1/q^2$, and one can use continued fractions to find such approximations, for instance $\pi \approx 22/7$. 

Metric diophantine approximation is about the typical rate of approximation. There are values of $\alpha$, such as the golden ratio, for which one can't do much better than Dirichlet's theorem. However, for all $\alpha$ away from a set of Lebesgue measure zero, one can beat it by a factor of $\log q$ and more. Khintchine's theorem is prototypical, asserting that if $\psi: \mathbb N \to [0, \infty)$ is decreasing then \[ \mathrm{meas} \{ \alpha \in [0,1]: \exists^\infty (q,a) \in \mathbb N \times \mathbb Z \quad | \alpha - a/q| < \psi(q)/q \} = \begin{cases} 1, & \text{if } \sum_{q=1}^\infty \psi(q) = \infty \\ 0,&\text{if } \sum_{q=1}^\infty \psi(q) < \infty. \end{cases} \] One can prove these sorts of results using the Borel-Cantelli lemmas, from probability theory: making a ball of radius $\psi(q)/q$ around each $a/q$, and grouping together the ones with the same $q$, the idea is to show that pairs of groups overlap more or less independently.

According to my mathematical upbringing, all phenomena are explained by the dichotomy between structure and randomness: either there is structure present, or else there is (pseudo)randomness. The probabilistic considerations above had initially led me to believe that randomness was the key to understanding metric diophantine approximation, but after working in the area for a while my opinion is closer to the opposite! The denominators of the good approximations to $\alpha$ lie in Bohr sets (after Harald Bohr, brother of the eminent physicist Niels Bohr) \[ B_N(\alpha, \delta) := \{ n \le N: \| n \alpha \| \le \delta \} \subset \mathbb N, \] where $\| \cdot \|$ denotes distance to the nearest integer. A central tenet of additive combinatorics is that Bohr sets look like generalised arithmetic progressions (GAPs).

I built the GAPs using continued fractions, enabling me to make progress towards the infamous Littlewood (c. 1930) and Duffin-Schaeffer (1941) conjectures. The former is about approximating two numbers at once in a multiplicative sense, that is to find approximations $a/q, b/q$ to $\alpha,\beta$ for which \[ \Bigl | \alpha - \frac a q \Bigr | \cdot \Bigl |\beta - \frac b q \Bigr| < \frac {10^{-100}}{q^3}, \] and the latter is about approximation by reduced fractions. With Niclas Technau, we have since developed a higher-dimensional structural theory using the geometry of numbers. Going forward, I hope to establish a Khintchine-type law for multiplicative approximation on planar curves.

Sam Chow, Oxford Mathematics

Monday, 29 October 2018

Nick Trefethen awarded honorary degrees by Fribourg and Stellenbosch Universities

Oxford Mathematician Professor Nick Trefethen, Professor of Numerical Analysis and Head of Oxford's Numerical Analysis Group has been awarded honorary degrees by the University of Fribourg in Switzerland and Stellenbosch University in South Africa where Nick was cited for his work in helping to cultivate a new generation of mathematical scientists on the African continent.

Nick's research spans a wide range within numerical analysis and applied mathematics, in particular the numerical solution of differential equations, fluid mechanics and numerical linear algebra. He is also the author of several very successful books which, as the Fribourg award acknowledges, have widened interest and nourished scientific discussion well beyond mathematics. 

Thursday, 25 October 2018

The Geometry of Differential Equations - Oxford Mathematics Research by Paul Tod

If you type fundamental anagram of calculus into Google you will be led eventually to the string of symbols 6accdæ13eff7i3l9n4o4qrr4s8t12ux, probably accompanied by an explanation more or less as follows: this is a recipe for an anagram - take six copies of a, two of c, one of d, one of æ and so on, then rearrange these letters into a chunk of Latin. The accepted solution is Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa although this has nine copies of t rather than eight, and this lack of a t is assumed to be an error of the author, who was Isaac Newton and who recorded the solution in a notebook. The string occurs in a letter of Newton from 1676, sent to the mathematician Leibniz via Henry Oldenburg, who was the first Secretary of the Royal Society and a much-used conduit for savants. Newton was seeking to establish priority in the invention of calculus without giving anything away. The string is preceded in Newton's letter by the sentence "...[B]ecause I cannot proceed with the explanation of it now, I have preferred to conceal it thus:". In other words he meant to hide his knowledge but still establish priority. Online translations give the meaning of the Latin as Given an equation involving any number of fluent quantities to find the fluxions, and vice versa, which would suggest it is disguising the fundamental theorem of calculus, 'fluents' and 'fluxions' being Newton's terms for time-varying quantities and their derivatives, but one can find looser translations online, along the lines of Differential equations rock or rule or are the way to go, and I'm voting for one of those here.

Newton was a great geometer and phrased his Principia Mathematica in the language of geometry, when a greater use of differential equations might have made it more accessible. Newton would probably have been pleased (if he was ever pleased) to see geometry re-emerging from the study of differential equations and that is what I want to describe here. Given an ordinary differential equation of some order, say $N$, and written as \begin{equation}\label{1} \frac{d^Ny}{dx^N}=F\left(x,y,\frac{dy}{dx},\ldots,\frac{d^{N-1}y}{dx^{N-1}}\right), \;\;\;\;\;\;\;\;\;(1)\end{equation} one can contemplate the set of all possible solutions. Call this $M$ (for moduli space, a common usage) then $M$ has $N$ dimensions, as one sees as follows: choose a value of $x$, say zero for simplicity, and specify the $N$ numbers $\left(y(0),dy/dx|_0,\ldots,d^{N-1}y/dx^{N-1}|_0\right)$, where the notation '$|_0$' should be read as 'evaluated at $x=0$', then, subject to reasonable conditions on $F$, there is one and only one solution of (1) with these values. Thus a point of $M$ is specified by $N$ numbers, which is what you mean by '$M$ having $N$ dimensions'.

If the function $F$ is linear, that is of the form \[F(x,y_0,y_1,\ldots,y_{N-1})=\alpha_0(x)y_0+\alpha_1(x)y_1+\ldots+\alpha_{N-1}(x)y_{N-1}\] then $M$ is a vector space and, for our purposes, flat and dull; but if $F$ is nonlinear then $M$ can be quite interesting.

I'll illustrate this with an example that goes back to the French mathematician Elie Cartan in 1943, [1], but which had a new lease of life from the 1980s, (see e.g. [2], [3]). Take $N=3$, so we are interested in nonlinear, third-order differential equations with corresponding three-dimensional moduli space $M$ of solutions. One thinks of $M$ as a manifold, imagine something like a smooth surface in some higher-dimensional Euclidean space, on which one can move around smoothly. Can one give $M$ a conformal metric? This is a rule for measuring angles between directions at any point, and is a slightly weaker notion than measuring lengths of vectors at any point (an angle being essentially obtained from a ratio of lengths), and it turns out that one can provided $F$ satisfies a certain condition. The condition is a bit messy to write down, so I won't, but really it has to be because it must still hold if we make new choices of $x$ and $y$ according to the change \[(x,y)\rightarrow(\hat{x},\hat{y})=(f(x,y),g(x,y)),\] for any $f$ or $g$, which certainly makes a big change in $F$, but not much change in M (for the expert: there is more freedom than this, of contact transformations rather than just point transformations).

Now one asks another question: can one also give $M$ a conformal connection (which can be thought of as a way of moving angles around and comparing them at different places)? Again there is a single condition on $F$ that allows this, and the upshot, noticed by Cartan, is that given these two conditions on $F$, the moduli space $M$ of solutions of $F$, with its conformal metric and connection, automatically satisfies the most natural conditions in three-dimensions which generalise the Einstein equations of general relativity in four-dimensions. This is really like a-rabbit-out-of-a-hat, and is typical of what happens in this study of the geometry of differential equations.

My second example is more complicated to describe - things get harder quickly now. Here we're following [4] but slightly rephrased as in [5], [6]. Consider then the following quite specific fifth-order differential equation: \begin{equation}\label{2} \left(\frac{d^2y}{dx^2}\right)^{\!2}\frac{d^5y}{dx^5}+\frac{40}{9}\left(\frac{d^3y}{dx^3}\right)^{\!3}-5\frac{d^2y}{dx^2}\frac{d^3y}{dx^3}\frac{d^4y}{dx^4}=0.\;\;\;\;\;\;\;\;(2) \end{equation} Of course this can be rearranged to look like (1) but I've written it like this to avoid denominators. It isn't too hard to solve (2), especially if you notice first that it is equivalent to \[\frac{d^3}{dx^3}\left[\left(\frac{d^2y}{dx^2}\right)^{\!-2/3}\right]=0.\] One rapidly finds that the general solution can be written implicitly as \begin{equation}\label{3}(\begin{array}{ccc} x & y & 1\\ \end{array})\left(\begin{array}{ccc} a_1&a_2&a_3\\ a_2&a_4&a_5\\ a_3&a_5&a_6\\ \end{array}\right)\left(\begin{array}{c}x\\ y\\ 1\\ \end{array}\right)=0, \;\;\;\;\;\;\;\;(3)\end{equation} where $a_1,\ldots,a_6$ are constants (there are six of these and we expect the solution to depend on only five constants, but there is freedom to rescale all six by yet another constant, which reduces the six to an effective five). For convenience we'll allow all quantities $x,y,a_1,\ldots,a_6$ to be complex now, and then (3) can be interpreted as a conic in the complex projective space ${\mathbb{CP}}^2$ (this is like a familiar conic in two-dimensions but complexified and with points added 'at infinity'). Thus $M$, the moduli space of solutions of the fifth-order differential equation (2) can be regarded as the five-complex-dimensional space of such conics. This $M$ is a symmetric space and as such has a metric satisfying the Einstein equations in this dimension. We can say a bit more about this metric: a vector at a point $p\in M$ can be thought of as two infinitesimally-separated conics (one at each end of the vector, thinking of the vector as a tiny arrow), and two conics meet in four points (at least in the complex and with points at infinity included). Thus a vector tangent to $M$ factorises into a product of four two-component 'vectors' (which are in fact spinors). In an index notation a vector $V$ can be written in terms of its components as \[V^a=V^{ABCD}=\alpha^{(A}\beta^B\gamma^C\delta^{D)},\] where $a$ runs from 0 to 4, $A,B,C,\ldots$ from 0 to 1, $\alpha,\beta,\gamma,\delta$ correspond to the four intersections of the infinitesimally-separated conics, and the round brackets imply symmetrisation of the indices contained. In a corresponding way the metric or Levi-Civita covariant derivative factorises: \[\nabla_a=\nabla_{ABCD}.\]Consequently the (linear, second-order) differential equation \begin{equation}\label{4}\nabla_{PQ(AB}\nabla_{CD)}^{\;\;\;\;PQ}\Phi=\lambda\nabla_{ABCD}\Phi,\;\;\;\;\;\;\;\;(4)\end{equation} for a scalar function $\Phi$ on $M$ and a constant $\lambda$ makes sense (the spinor indices are raised or lowered with the spinor 'metric' $\epsilon_{AB}$ or $\epsilon^{AB}$ from which the metric of $M$ can be constructed).

Now here comes the rabbit out of the hat: one fixes a specific $\lambda$ and chooses a solution $\Phi$ of (4) with a non-degeneracy property \begin{equation}\label{5}\Phi_{ABCD}\Phi^{AB}_{\;\;\;PQ}\Phi^{CDPQ}\neq0\mbox{ on }\Phi=0,\;\;\;\;\;\;\;\;(5)\end{equation} (and here $\Phi_{ABCD}=\nabla_{ABCD}\Phi$); we are going to define a new metric on the four-dimensional surface $N=\{\Phi=0\}\subset M$ and we accomplish this by choosing two linearly independent solutions $h_{(1)}^{ABC},h_{(2)}^{ABC}$ of \[\Phi_{ABCD}h_{(i)}^{BCD}=0\mbox{ on }\Phi=0,\] an equation which has a two-dimensional vector space of solutions by virtue of the nondegeneracy condition (5) that we've imposed; then one carefully chooses a linearly independent pair of spinors (or spinor dyad) $(o^A,\iota^A)$ for $M$ and constructs the set of four linearly-independent vectors (or tetrad) \[e_1=o^{(A}h_{(1)}^{BCD)},\;e_2=\iota^{(A}h_{(1)}^{BCD)},\;e_3=o^{(A}h_{(2)}^{BCD)},\;e_4=\iota^{(A}h_{(2)}^{BCD)};\] this is a tetrad of vectors tangent to $N$ and determines a metric on $N$ by \[g=e_1\odot e_4-e_2\odot e_3.\] The rabbit is that this metric automatically has special curvature, in fact it has anti-self-dual Weyl curvature, and a bit more besides: see [5] for details.

For more on Paul's work click here.

[1] E. Cartan, Sur une classe d'espaces de Weyl, Annales scientifiques de l'E.N.S. 3e série, tome 60 (1943), 1-16

[2] N.J.Hitchin, Complex manifolds and Einstein's equations. Twistor geometry and nonlinear systems (Primorsko, 1980), 73-99, Lecture Notes in Math., 970, Springer, Berlin-New York, 1982.

[3] P.Tod, Einstein-Weyl spaces and third-order differential equations. J. Math. Phys. 41 (2000) 5572-5581.

[4] D Moraru, A new construction of anti-self-dual four-manifolds. Ann. Glob. Anal. Geom. 38, (2010), 77-92.

[5] M. Dunajski and P. Tod, Conics, Twistors, and anti-self-dual tri-Kahler metrics

[6] M. Dunajski and P. Tod, An example of the geometry of a 5th-order ODE: the metric on the space of conics 

Thursday, 25 October 2018

An Introduction to Complex Numbers - Oxford Mathematics first year student lecture now online for the first time

Much is written about life as an undergraduate at Oxford but what is it really like? As Oxford Mathematics's new first-year students arrive (273 of them, comprising 33 nationalities) we thought we would take the opportunity to go behind the scenes and share some of their experiences.

Our starting point is a first week lecture. In this case the second lecture from 'An Introduction to Complex Numbers' by Dr. Vicky Neale. Whether you are a past student, an aspiring student or just curious as to how teaching works, come and take a seat. We have already featured snippets from the lecture on social media where comments have ranged from a debate about whiteboards to discussions abut standards. However, there has also been appreciation of the fact that we are giving an insight in to a system that is sometimes seem as unnecessarily mysterious. In fact there is no mystery, just an opportunity to see how we present the subject and how that differs from the school experience, as much in presentation as content though of course that stiffens as the weeks go by.

So take your seat and let us know what you think.



Thursday, 25 October 2018

Bach and the Cosmos - James Sparks and City of London Sinfonia. Oxford Mathematics Public Lecture now online

According to John Eliot Gardiner in his biography of Johann Sebastian Bach, nothing in Bach's rigid Lutheran schooling explains the scientific precision of his work. However, that precision has attracted scientists and mathematicians in particular to the composer's work, not least as its search for structure and beauty seems to chime with their own approach to their subject.

In this Oxford Mathematics Public Lecture Oxford Mathematician James Sparks, himself a former organ scholar at Selwyn College Cambridge, demonstrates just how explicit Bach's mathematical framing is and City of London Sinfonia elucidate with excerpts from the Goldberg Variations. This was one of our most successful Public Lectures, an evening where the Sciences and the Humanities really were in harmony.

Please note this film does not include the full concert performance of the Goldberg Variations.