Monday, 16 January 2017

The magic of numbers - finding structure in randomness

Mathematics is full of challenges that remain unanswered. The field of Number Theory is home to some of the most intense and fascinating work. Two Oxford mathematicians, Ben Green and Tom Sanders, have recently made an important breakthrough in an especially tantalising problem relating to arithmetic structure within the whole numbers.

Imagine colouring every positive whole number with one of three colours, say red, green and blue. You might end up with this colouring: R1, G2, B3, B4, G5, R6, B7, R8, R9, G10, B11, B12, R13, G14, B15, B16, R17 ...

Or perhaps you chose one of the other infinitely many possibilities. Can you always pick two of these numbers, say x and y, so that x, y, x+y and xy all have the same colour?

For example, in our choice of colouring above we see that 3, 4, 3+4=7 and 3x4=12 are all blue, so it's possible for this colouring.  But will we always be able to do this, regardless of the colouring? And what if we use four colours, or even more, will we always be able to do it then too?

This is a well-known question in Ramsey theory, a branch of combinatorics that seeks to establish the existence of structure (such as the pattern x, y, x+y and xy all having the same colour) in randomness (such as all the many colourings). Results in Ramsey theory link with other areas of mathematics, and also have applications in other fields, notably in theoretical computer science.

A hundred years ago, the mathematician Issai Schur showed that if we colour every positive whole number with one of three or more colours then there are always two numbers, say x and y, so that x, y and x+y all have the same colour.  Extending this to handle multiplication as well as addition has turned out to be a significant challenge.

Ben and Tom have solved a sort of 'model' problem, where instead of colouring the integers they work with a finite analogue. This approach of considering a model problem has proved extremely fruitful for a number of other related questions, but it was far from clear how to proceed in this case. Ben and Tom introduced a number of new techniques in order to solve the problem in this finite model situation.

How does the finite analogue work?  Instead of colouring the positive whole numbers, Green and Sanders use a different system that shares similar arithmetic properties. Imagine a clock that shows 7 hours (rather than the familiar 12).  Every 7 hours, the hour hand gets back round to the top.  We can do arithmetic in this scenario too, for example 13 + 17 = 6 + 3 = 2 in this world, and 13 x 17 = 6 x 3 = 4. Ben and Tom showed that if the 7 values in this system are coloured using three or more colours then there must be many x and y for which x, y, x+y and xy all have the same colour, and similarly for any such system where the number of values is prime. Their work has been published in the new journal Discrete Analysis.

Indeed there has already been follow-up work, with Ben and his Oxford Graduate Student Sofia Lindqvist using similar ideas to resolve a question about monochromatic solutions to the equation $x+y=z^2$.

The original problem may remain unanswered, but mathematicians now have a way in and are working to establish whether Ben and Tom’s work provides the answer, both to this and maybe further problems in number theory. As so often, successes are incremental and hard-fought as mathematicians continue to map the structures that populate their and our world.

Wednesday, 11 January 2017

Stephen Hawking's Oxford Mathematics Public Lecture - live podcast CANCELLED



In recognition of a lifetime's contribution across the mathematical sciences, we are initiating a series of annual Public Lectures in honour of Roger Penrose. The first lecture will be given by his long-time collaborator and friend Stephen Hawking on 18th January at 5pm GMT. The lecture is sold out, but we will be podcasting live

Stephen Hawking is the former Lucasian Professor of Mathematics at the University of Cambridge and now the Dennis Stanton Avery and Sally Tsui Wong-Avery Director of Research at the Department of Applied Mathematics and Theoretical Physics and Founder of the Centre for Theoretical Cosmology at Cambridge.


Friday, 6 January 2017

The Mathematics of Visual Illusions - Christmas lecture online

Puzzling things happen in human perception when ambiguous or incomplete information is presented to the eyes. For example, illusions, or multistable figures occur when a single image can be perceived in several ways. 

In the Oxford Mathematics Christmas Public Lecture Ian Stewart demonstrates how these phenomena provide clues about the workings of the visual system, with reference to recent research which has modelled simplified, systematic methods by which the brain can make decisions.

Ian Stewart is Emeritus Professor of Mathematics in the University of Warwick.





Tuesday, 3 January 2017

18th Century Oxford Mathematics - Halley to Hornsby

In our final series of Oxford Mathematics History Posters we look at Oxford’s role in the development of Newtonian philosophy in the 18th Century. In particular we focus on Edmond Halley, the most famous English astronomer of his day and Savilian Professor of Geometry, and Thomas Hornsby, Sedleian Professor of Natural Philosophy and founder of the Radcliffe Observatory which appropriately now sits close to the new Mathematical Institute.

PDF icon Halley to Hornsby.pdf

Tuesday, 3 January 2017

The mathematics of violent plastic deformation

This picture shows the "Z" machine at Sandia Labs in New Mexico producing, for a tiny fraction of a second, 290 TW of power - about 100 times the average electricity consumption of the entire planet. This astonishing power is used to subject metal samples to enormous pressures up to 10 million atmospheres, causing them to undergo violent plastic deformation at velocities up to 10 km/s. How should such extreme behaviour be described mathematically?

Oxford DPhil student Stuart Thomson is working with Peter Howell, John Ockendon, Hilary Ockendon and collaborators at AWE to answer this question. To a first approximation, the plastically flowing metal behaves like a compressible inviscid gas, with small but important elastic waves superimposed. The team’s simulations and analysis explain and quantify the experimentally observed behaviour, and reveal a fascinating and previously unexplored phenomenon whereby fast-moving elastic waves reflect off slower-moving plastic waves. The results shed crucial light on the inverse problem of backing out the effective equation of state from the experimentally measured response of the sample, as well as posing fundamental theoretical questions about singularly perturbed hyperbolic systems.

This research is funded by an EPSRC Industrial CASE award through the Smith Institute for Industrial Mathematics and System Engineering.

Wednesday, 14 December 2016

The Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjectures

As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician Gui-Qiang G Chen discusses his work on the Mathematics of Shock Reflection-Diffraction.

Shock waves are fundamental in nature, especially in high-speed fluid flows. Shocks are often generated by supersonic or near-sonic aircraft, explosions, solar wind, and other natural processes. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws - nonlinear partial differential equations (PDEs) of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns as conjectured by von Neumann (1943), it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science.

Oxford mathematician Gui-Qiang G. Chen and his collaborator Mikhail Feldman (University of Wisconsin-Madison) have introduced new ideas and developed techniques for solving fundamental open problems for multidimensional (M-D) shock reflection-diffraction and related free boundary problems for nonlinear conservation laws of mixed hyperbolic-elliptic type in a series of their papers. In particular, in their Annals paper, they developed the first mathematical approach to the global problem of shock reflection-diffraction by wedges and employed the approach to solve rigorously the problem with large-angle wedges for potential flow through careful mathematical analysis. This paper was awarded the Analysis of Partial Differential Equations Prize in 2011 by the Society for Industrial and Applied Mathematics. 

In the last five years, further significant advances have been made, including their complete solution to von Neumann’s sonic conjecture and detachment conjecture for potential flow. These are reported in their forthcoming research monograph published in the Princeton Series in Annals of Mathematics Studies. This monograph offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of their original mathematical proofs of von Neumann's conjectures, and a collection of related results and new techniques in the analysis of PDEs, as well as a set of fundamental open problems for further development. The approaches and techniques that Chen and his collaborators have developed will be useful in solving nonlinear problems with similar difficulties and open up new research directions.

Wednesday, 30 November 2016

Levelling the access playing-field

Oxford Mathematics and Imperial College have joined forces to co-pilot a new programme aimed at levelling the playing-field for bright young mathematicians.

The two universities use the Maths Admission Test (MAT) as the basis for undergraduate assessment. It’s meant to be fair to all, particularly as it is not based on the Further Maths syllabus which many schools do not offer. But is it? How can it take in to account all the other factors that determine a candidate’s preparedness for such a potentially daunting challenge?

The Problem Solving MATters programme is designed to prepare students from less advantaged backgrounds for achieving success in the MAT. It comprises three face-to-face study days, focussing on specific problem-solving skills, with a short practice exam in the final session; three summer assignments to further develop thinking skills and technique; and five online follow-up sessions, designed to consolidate new skills in the run up to the MAT itself. Crucially, participants are supported by student mentors who offer feedback throughout the process.

The course has been made possible by the generosity of Oxford Mathematics Alumnus Tony Hill.

“My aim”, says Tony, "is that the programme will continue and be rolled out to other Russell Group Universities, so we can get the best people into Maths departments, not just the best-prepared. This programme gives young people from less advantaged backgrounds an opportunity to see what Imperial and Oxford are actually like. As well as being taught by experts and mentored by undergraduates, they have a chance to look around, see people like them from all over the country and to visualise themselves in such a place."

Tony himself grew up on a council estate and was the first from his family to go to university. He understands the issues and he's passionate about helping talented young people overcome common stumbling blocks, in particular "those kids from lower socio-economic backgrounds or whose families don't value education; those whose school isn't very good generally or at teaching Maths, or where they have the attitude of kids from round here don't go to that type of university...Compare that to a kid coming from a good school that's strong in maths and with a strong tradition of getting their students into good universities. In one sense it's equal and in another, it's not."

If you would like to know more about the courses for 2017 please email

Thanks to Jean Bywater at Imperial College for researching and writing the original article.


Tuesday, 29 November 2016

Improving the quality and safety of x-rays

X-ray imaging is an important technique for a variety of applications including medical imaging, industrial inspection and airport security. An X-ray image shows a two-dimensional projection of a three-dimensional body. The original 3D information can be recovered if multiple images are given of the same object from different viewpoints. The process of recovering 3D information from a set of 2D X-ray projections is called Computed Tomography (CT).

Traditional CT scanners are based on a single moving source, rotating around the object to reconstruct. In this set-up, images are taken sequentially and the resulting reconstruction problem gives rise to a linear system of equations.

The innovation of X-ray emitter arrays allows for a novel type of X-ray scanning device with faster image acquisition due to multiple simultaneously emitting sources. Acquisition speed is an important factor in medical imaging because it can help to avoid artefacts from motion of the analysed tissue. Another major advantage of emitter arrays is that they result in light weight, highly mobile 3D imaging systems that can be taken to the patient rather than having to move the patient to a radiology suite.

However, two or more sources emitting simultaneously can yield measurements from spatially and temporally overlapping rays. This imposes a new type of image reconstruction problem based on nonlinear constraints that traditional linear image reconstruction methods cannot cope with.

Oxford Mathematicians Raphael Hauser and Maria Klodt have derived a mathematical model for this new type of image reconstruction problem, and developed a reconstruction method that allows the recovery of images from measurements with overlapping rays. Based on compressed sensing, the method allows for reconstruction from undersampled data, which means that the number of reconstructed densities is higher than the number of measurements, which enables reduced doses (see the full paper).

The method has been successfully applied to real X-ray measurements in cooperation with Gil Travish and Paul Betteridge from industrial partner Adaptix, an Oxford-based start-up company. The Adaptix scanners acquire images with a flat panel of comparatively small emitters with small opening angles of the emitter cones, arranged in a fixed grid which can allow for small devices with reduced doses.

The new image reconstruction method opens new possibilities for X-ray scanner design, because it allows for a new class of hand-held X-ray scanning devices, where emitter and detector positions cannot be aligned exactly, and overlapping of emitter cones cannot be avoided.

Friday, 25 November 2016

Maria Bruna wins the Women of the Future Science award

Oxford Mathematician Maria Bruna has won the Women of the Future Science award. The Women of the Future Awards, founded by Pinky Lilani in 2006, were conceived to provide a platform for the pipeline of female talent in the UK. Now in their 11th year they recognise the inspirational young female stars of today and tomorrow. They are open to women aged 35 or under and celebrate talent across categories including business, culture, media, technology and more.

Maria's work focuses on partial differential equations, stochastic simulation algorithms and the application of these techniques to the modelling of biological and ecological systems. 

Thursday, 24 November 2016

Random Walks 3 – The beauty and symmetry of ancient tiles

Beauty is in the eye of the beholder, but what about symmetry? In our final feature on mathematicians let loose in the Ashmolean Museum, Oxford Mathematician Balázs Szendrői investigates the beauty of symmetry in the Museum's Islamic art works. As he explains, no matter what the tile pattern may look like, its underlying symmetry configuration belongs to a small set of possibilities. 

If you are interested in the Random Walks series have a look at the previous films.