Friday, 17 January 2020

Interested in Graduate Study but not sure you can make it work? UNIQ+ might just help

Financial, socio-economic and other life circumstances can make it difficult for some to continue studying beyond an undergraduate degree. UNIQ+ is intended to encourage access to postgraduate study from talented undergraduates from across the UK who would find continuing into postgraduate study a challenge for reasons other than their academic ability.

The programme offers paid summer research internships giving talented UK undergraduate students the opportunity to discover what postgraduate study is like at Oxford through research experience in the University’s state-of-the-art facilities, working alongside our students and staff.

The 2020 programmes will run from 6 July for seven weeks.

Applications are now open. The deadline is 12 noon on Monday 24 February 2020.

For full information, including eligibility criteria, click here.

Tuesday, 14 January 2020

Nick Trefethen awarded the 2020 John von Neumann Prize by SIAM

Oxford Mathematician Nick Trefethen is the 2020 recipient of the John von Neumann Prize, the highest honour and flagship lecture of Society for Industrial and Applied Mathematics (SIAM), in recognition of his ground-breaking contributions across many areas of numerical analysis. 

SIAM awards the John von Neumann Prize annually to an individual for outstanding and distinguished contributions to the field of applied mathematics and for the effective communication of these ideas to the community. It is one of SIAM’s most distinguished prizes as well as an important lecture at the SIAM Annual Meeting. The selection committee states, “He is an outstanding expositor of applied mathematics and his books are beautifully written, widely accessible, and highly original.”

The John von Neumann Lecture was established in 1959 to honor von Neumann, a Hungarian-American mathematician, physicist, and computer scientist, whose seminal work helped lead to the founding of modern computing. 

Nick is Professor of Numerical Analysis in Oxford, a Fellow of Balliol College and Head of Oxford Mathematics's Numerical Analysis Group. He has published around 140 journal papers spanning a wide range of areas within numerical analysis and applied mathematics, including non-normal eigenvalue problems and applications, spectral methods for differential equations, numerical linear algebra, fluid mechanics, computational complex analysis, and approximation theory.

Nick will deliver this flagship lecture at the Second Joint SIAM/CAIMS Annual Meeting (AN20) in July.

Friday, 3 January 2020

Modelling functions of sequential data with neural networks and the signature transform

Oxford Mathematician Patrick Kidger talks about his recent work on applying the tools of controlled differential equations to machine learning.

Sequential Data

The changing air pressure at a particular location may be thought of as a sequence in $\mathbb{R}$; the motion of a pen on paper may be thought of as a sequence in $\mathbb{R}^2$; the changes within financial markets may be thought of as a sequence in $\mathbb{R}^d$, with $d$ potentially very large.

The goal is often to learn some function of this data, for example to understand the weather, to classify what letter has been drawn, or to predict how financial markets will change.

In all of these cases, the data is ordered sequentially, meaning that it comes with a natural path-like structure: in general the data may be thought of as a discretisation of a path $f \colon [0, 1] \to V$, where $V$ is some Banach space. (In practice this is typically $V = \mathbb{R}^d$.)

The Signature Transform

When we know that data comes with some extra structure like this, we can seek to exploit that knowledge by using tools specifically adapated to the problem. For example, a tool for sequential data that is familiar to many people is the Fourier transform.

Here we use something similar, called the signature transform, which is famous for its use in rough path theory and controlled differential equations.

The signature transform has a rather complicated looking definition: \[ \mathrm{Sig}^N(f) = \left(\left(\,\underset{0 < t_1 < \cdots < t_k < 1}{\int \cdots \int} \prod_{j = 1}^k \frac{\mathrm{d}f_{i_j}}{\mathrm{d}t}(t_j)\mathrm{d}t_1\cdots\mathrm{d}t_k \right)_{1\leq i_1,\ldots, i_k \leq d}\right)_{1\leq k \leq N} \]

Whilst the Fourier transform extracts information about frequency, the signature transform instead extracts information about order and area. (It turns out that order and area are, in a certain sense, the same thing.)

Furthermore (and unlike the Fourier transform), order and area represent all possible nonlinear effects: the signature transform is a universal nonlinearity, meaning that every continuous function of the underlying path corresponds to just a linear function of its signature.

(Technically speaking, this is because the Fourier transform uses a basis for the space of paths, whilst the signature transform uses a basis for the space of functions of paths.)

Besides this, the signature transform has many other nice properties, such as robustness to missing or irregularly sampled data, optional translation invariance, and optional sampling invariance.

Applications to Machine Learning

Machine learning, and in particular neural networks, is famous for its many recent achievements, from image classification to self driving cars.

Given the great theoretical success of the signature transform, and the great empirical success of neural networks, it has been natural to try and bring these two together.

In particular, the problem of choosing activation functions and pooling functions for neural networks has usually been a matter of heuristics. Here, however, the theory behind the signature transform makes it a mathematically well-motivated choice of pooling function, specifically adapted to handle sequential data such as time series.

Bringing these two points of view together has been the purpose of the recent paper Deep Signature Transforms (accepted at NeurIPS 2019) by Patrick Kidger, Patric Bonnier, Imanol Perez Arribas, Cristopher Salvi, and Terry Lyons. Alongside this we have released Signatory, an efficient implementation of the signature transform capable of integrating with modern deep learning frameworks.

Thursday, 2 January 2020

Using modular forms to construct points on elliptic curves

Oxford Mathematician Alan Lauder works on elliptic curves and modular forms, and methods for constructing points on the former using the latter. Read more here.

"An elliptic curve is described by an equation of the form \[ E: Y^2 = X^3 + a X + b \] where $a,b \in \mathbb{Z}$ with $4 a^3 + 27 b^2 \ne 0$. One calls solutions $(x,y)$ to this equation points on the elliptic curve $E$. There is an elegant geometric method of "adding'' two points $(x,y),(x^\prime, y^\prime)$ on $E$ to obtain a third point. The set $E(\mathbb{Q})$ of all rational points $(x,y) \in \mathbb{Q}^2$ on $E$, along with a point at infinity, forms a finitely generated abelian group under this operation of addition. One has then \[ E(\mathbb{Q}) \cong \mathbb{Z}^{r(E)} \oplus T \] where $T$ is the finite torsion group and $r(E)$ is called the rank of $E$. Note that $E$ has an infinite number of rational points exactly when the rank is positive.

One expects that the rank is positive around half the time (this expectation is a consequence of the Birch and Swinnerton-Dyer conjecture). So if one writes down an elliptic curve at random, rather remarkably around half the time it will have an infinite number of rational points on it. The difficulty though is to find these points, or rather, some points of infinite order which generate the group $E(\mathbb{Q})$ modulo torsion.

To try and do this, one associates to $E$ a seemingly unrelated object $f_E$ called a modular form of weight $2$. Using this association, there is a complex analytic construction which often allows one to quickly find a point $P$ on $E$ of infinite order. This method of Heegner points was developed in Oxford by Birch in the 1960s and 70s, following the pioneering work of the German school teacher Heegner in the 1950s. The Heegner point $P = (x,y)$ naturally has coordinates in some imaginary quadratic field $K$, but one can choose the field $K$ appropriately to force $P$ to be a rational point.

To an imaginary quadratic field $K$ one can also associate a modular form, but this time of weight $1$. In my work with Henri Darmon and Victor Rotger, using this association we found a new formula for Heegner points. The modular forms of weight 1 which arise from imaginary quadratic fields are of a special form, and there are many modular forms of weight 1 which do not arise in this way. Our new formula still works (conjecturally) for these more general modular forms, but in this case defines points on $E$ over other number fields; namely, over class fields of real quadratic fields, and also fields with Galois group $A_4, S_4$ and $A_5$. The points over class fields of real quadratic fields were first discovered by Darmon twenty years ago using a completely different approach. Our formula gives a unifying perspective on the methods of Heegner and Darmon, as well as defining points in new settings.''

For more on the work click here.

Friday, 27 December 2019

Nick Woodhouse appointed CBE in 2020 New Year Honours List

Professor Nick Woodhouse, Emeritus Professor of Mathematics in Oxford and Emeritus Fellow of Wadham College, former Head of the Mathematical Institute and previously President of the Clay Mathematics Institute has been appointed CBE in the 2020 New Year Honours List for services to mathematics.

Nick has had a distinguished career as both a researcher and a leading administrator in the University. His research has been at the interface between mathematics and physics, initially in relativity, and later in more general connections between geometry and physical theory, notably via twistor theory.  In parallel he led the Mathematical Institute in Oxford at a time of major expansion and was the leading figure in the Institute's move to the Andrew Wiles Building, completed in 2013. His time as President of the Clay Mathematics Institute saw its profile and influence increase and its roster of talented Clay Research Fellows grow.

Nick also played a leading role in the administration of the wider University including a period as Deputy Head of the Mathematical, Physical and Life Sciences Division; and was a member of the North Commission set up in 1997 to review the management and structure of the collegiate University and whose recomendations helped shape Oxford as it operates in 2020.

Monday, 9 December 2019

The Penrose Proofs: an exhibition of Roger Penrose’s Scientific Drawings 1-6

As you might expect from a man whose family included the Surrealist artist Roland Penrose, Roger Penrose has always thought visually. That thinking is captured brilliantly in this selection of Roger’s drawings that he produced for his published works and papers.

From quasi-symmetric patterns to graphic illustrations of the paradoxical three versions of reality via twistor theory and the brain, this selection captures the stunning range of Roger’s scientific work and the visual thinking that inspires and describes it.

Mezzanine Level
Mathematical Institute

10 December 2019- 31 March 2020

Friday, 22 November 2019

Oxford Mathematics London Public Lecture with Tim Gowers and Hannah Fry now online

Oxford Mathematics London Public Lecture: Timothy Gowers - Productive generalization: one reason we will never run out of interesting mathematical questions

In our Oxford Mathematics London Public Lecture held at the Science Museum, Fields Medallist Tim Gowers uses the principle of generalization to show how mathematics progresses in its relentless pursuit of problems.

After the lecture in a fascinating Q&A with Hannah Fry, Tim discusses how he approaches problems, both mathematical and personal.

Oxford Mathematics Public Lectures are generously supported by XTX Markets. 




Friday, 22 November 2019

Oxford Mathematics 2nd Year Student Lecture on Quantum Theory now online

Our latest online student lecture is the first in the Quantum Theory course for Second Year Students. Fernando Alday reflects on the breakdown of the deterministic world and describes some of the experiments that defined the new Quantum Reality.

This is the sixth lecture in our series of Oxford Mathematics Student Lectures. The lectures aim to throw a light on the student experience and how we teach. All lectures are followed by tutorials where pairs of students spend an hour with their tutor to go through the lectures and accompanying work sheets.

An overview of the course and the relevant materials are available here:







Tuesday, 19 November 2019

Exploring wrinkling in thin membranes

For centuries, engineers have sought to prevent structures from buckling under heavy loads or large impacts, constructing ever larger buildings and safer vehicles. However, recent advances in soft matter are redefining the way we manipulate materials. In particular, an age-old aversion to buckling is being recast in a new light as researchers find that structural instabilities can be harnessed for functionality. This paradigm shift, from buckliphobia to buckliphilia, permits re-evaluation of the potential of soft, deformable structures, opening up methods of exploiting buckling to tune material characteristics or develop metamaterials.

Elastic instabilities provide a means of generating regular topographies with a well-defined wavelength. For example, a thin elastic film attached to a softer substrate buckles into an array of regular wrinkles under quasi-static compression. The wrinkle wavelength is selected by the mechanical properties of the system, so that different wavelengths are typically attained through variation of the film thickness. In an article recently published in the Proceedings of the National Academy of Sciences (PNAS) Oxford Mathematicians Finn Box, Doireann O’KielyOusmane Kodio, Maxime Inizan, and Dominic Vella and Alfonso A. Castrejón-Pita from Oxford's Fluid Dynamics Laboratory in the Department of Engineering Science show that, for a film of given thickness, variation in the wrinkle wavelength can instead be achieved via impact.

The researchers dropped steel spheres onto ultra-thin sheets of polystyrene, floating on water, and filmed what happened with a high-speed video camera. They found that ballistic impact caused the floating sheet to retract inwards, and the compression associated with this retraction induced buckling – resulting in a striking pattern of radial wrinkles.

Importantly, the distance between neighbouring wrinkles was found to evolve in time (equivalently, the number of wrinkles decreased in time) in stark contrast to previous observations in both static indentation experiments and previous dynamic impact experiments. Through mathematical modelling and systematic experimentation, the researchers found that the inertial response of the liquid substrate (i.e. the water on which the sheet floats) controlled the evolution of the wrinkle pattern.

This demonstration of wrinkle coarsening suggests that a dynamic substrate stiffness may provide a means of breaking away from the single, static wavelength that is selected by material properties alone, opening the route towards dynamic tuning of wrinkle-patterned topographies. This novel method for tunable wrinkle formation may prove to be a useful fabrication technique in a range of engineering applications that require regular, patterned topographies.

In their work, the researchers demonstrated that rapid coarsening of wrinkle wavelengths occurs with wavelengths on the order of 100 microns, making it readily observable. The uncovered mechanism for wrinkle formation is scale-independent, however, which indicates that this dynamic method of altering surface structure is suitable for reproduction at the nanoscale where the lengthscale of the wrinkles would be small enough for use in optical applications including photonic materials, which require periodic structures with period comparable with the wavelength of visible light. The reported dynamic wrinkling of thin, floating sheets is fast though, occurring within 10s of milliseconds – blink twice and you’ll miss it. 

A film showing the Dynamic wrinkling of thin sheets


Tuesday, 19 November 2019

How is the global energy challenge related to chaos and machine learning?

Energy production is arguably one of the most important factors underlying modern civilisation. Energy allows us to inhabit inhospitable parts of the Earth in relative comfort (using heating and air conditioning), create large cities (by efficiently transporting food and pumping water), or maintain our health (providing the energy for water purification). It also connects people by allowing long-distance travel and facilitating digital communication.

But energy is a sensitive subject at the moment, mainly for two reasons. Firstly, the way we currently produce energy is not sustainable: the Earth’s oil, coal, gas, and uranium reserves are finite, and we are tearing through them. Secondly, it is widely acknowledged that burning fossil fuels is affecting the Earth’s climate as we release greenhouse gases into the atmosphere. How we deal with these issues is a vital, but challenging problem.

Tokamaks are nuclear fusion reactors which are designed to prove the feasibility of fusion as a large-scale and carbon-free source of energy. These reactors are suggested as one of the potential solutions to the global energy challenge. Nuclear fusion involves controlling plasmas at temperatures of 100 Million degrees Celsius, which is ten times the temperature of the Sun. However, this produces unwanted turbulence in the tokamak due to the huge temperature gradients at certain plasma parameters. One of the challenges for Culham Centre for Fusion Energy (CCFE) is to identify such chaotic scenarios in order to avoid damage to the facility and to optimise the efficiency of energy production by stabilising the plasma.

Attracted by the recent spectacular successes of machine learning techniques for image classification, Debasmita Samaddar, a computational plasma physicist from CCFE, approached Oxford Mathematicians Nicolas Boullé, Vassilios Dallas, and Yuji Nakatsukasa to investigate whether machine learning can be employed to effectively control fusion reactors. The research that was carried out focused on how time series can be classified into chaotic or not (see Fig. 1) using machine learning.

Figure 1: A non-chaotic (left) and a chaotic (right) time series generated by the Lorenz system.

Contrary to standard machine learning techniques, the neural network was trained on a different and simpler set than the testing set of interest in order to demonstrate the generalisation ability of neural networks in this classification problem. The main challenge is to learn the chaotic features of the training set, without overfitting, and generalise on the testing data set, which behaves differently. Using a neural network that was trained on the Lorenz system, which is a system of three coupled nonlinear Ordinary Differential Equations (ODEs), we were able to classify time series of the Kuramoto-Sivashinsky (KS) equation  (see Fig. 2) as chaotic or not with high accuracy. The KS equation arises in a wide range of physical problems including instabilities in plasmas and is a characteristic example of a nonlinear PDE that exhibits spatiotemporal chaos.


Figure 2: A spatiotemporal chaotic solution of the Kuramoto-Sivashinsky equation (left) and its corresponding chaotic energy time series (right).

This important scientific result from this cross-disciplinary collaboration (facilitated by the Industrially Focused Mathematical Modelling Centre for Doctoral Training in Oxford) suggests that neural networks are able to identify the critical regimes that a fusion reactor might exhibit, paving the way to resolve central problems about the stability of CCFE's fusion reactors. It will be of great interest to see whether this work proves to be vital for the design of the next generation fusion reactors, helping them provide a sustainable energy solution.