Monday, 30 January 2017 
In our latest Public Lecture Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truthtelling.
For details and notification of the live podcast on 8 February at 4pm please click here.

Wednesday, 25 January 2017 
Organometal halide perovskite (OMHP) is hardly a household name, but this new material is the source of much interest, not least for Oxford Applied Mathematicians Victor Burlakov and Alain Goriely as they model the fabrication and operation of solar cells.
One of the main advantages of OMHP is that the thin films on its base can be produced in a very inexpensive way via a solution deposition with subsequent heat treatment at moderate temperatures. Victor and Alain developed a generic theoretical framework for calculating surface coverage by a solid film of material dewetting on the substrate. Using experimental data from OMHP thin films as an example, they calculated surface coverage for a wide range of annealing (heating) temperatures and film thicknesses. Their model accurately reproduced solutioncast thin film coverage (see the figure) and identified methods for both high and low levels of surface coverage.
Expanding their research Victor and Alain have also looked at the kinetics of photogenerated charge carriers in OMHP. By modelling the time decay of photoluminescence in the material, they extracted important information about charge carrier lifetime and concentration of intrinsic point defects. The latter are highly detrimental for the photovoltaic performance of OMHP. The models, together with the experimental studies of colleagues, clarified the origin of the point defects and consequently identified a means of significantly decreasing their concentration.
Victor and Alain’s research can be explored in more detail on Victor and Alain's homepages. Their work reiterates the central importance of mathematical modelling in addressing real world problems.

Thursday, 19 January 2017 
Think of a mathematician and you might imagine an isolated individual fueled by coffee whose immaculate if incomprehensible papers may, in the fullness of time, via a decadeslong dry chain of citations, be made use of by an industrialist (via one or two other dedicated mathematicians).
Not so, says new research by Oxford Mathematician and Computer Scientist Ursula Martin and evaluator Laura Meagher. Instead they reveal a vibrant and fertile environment where human interaction is the key.
Mathematics’ impact in every walk of life is astounding. Deloitte estimate that 10% of all UK jobs and 16% of total UK GDP is a direct result of mathematics. Ursula and Laura’s research puts the flesh on those figures, literally so as it demonstrates that mathematical impact is brought about above all by human interaction, long term relationships and close working with other disciplines and end users.
In the context of an increasing interest in generating and measuring impacts across the academic and funding worlds, Ursula and Laura used the trove of data provided by the 2014 Research Excellence Framework (which assesses the quality and impact of research across higher education in the UK) to dig down into 209 examples of the impact of UK mathematics and statistics.
Complementing this with surveys and indepth interviews, they identified a diverse ecosystem of people and ideas across mathematics, an ecosystem that includes the many other disciplines where mathematics is crucial as well as the many end users and beneficiaries of mathematical research. Their research also highlighted the role of specialist individuals in building long term relationships.
Moreover, the varieties of impact were striking, both the deep conceptual work that can reshape a whole field, and the detailed deployment of that work in a specific problem domain, both mathematical and beyond.
Finally, their work reinforced the crucial role of universities in developing a culture supportive of impact generation which reinforces the distinctive but allpervasive nature of mathematics, a discipline that is underpinning and influencing so many of the scientific, technological and social questions we are asking of our world.

Wednesday, 18 January 2017 
In our latest Public Lecture Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truthtelling.
8 February, 4pm, Mathematical Institute, Oxford. Please email externalrelations@maths.ox.ac.uk to register

Monday, 16 January 2017 
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 wellknown 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 followup 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 hardfought as mathematicians continue to map the structures that populate their and our world.

Wednesday, 11 January 2017 
UNFORTUNATELY THIS HAS BEEN CANCELLED. A NEW DATE WILL BE SET SOON.

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 longtime 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 WongAvery 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 
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 
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.
Halley to Hornsby.pdf

Tuesday, 3 January 2017 
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 fastmoving elastic waves reflect off slowermoving 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 
As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician GuiQiang G Chen discusses his work on the Mathematics of Shock ReflectionDiffraction.
Shock waves are fundamental in nature, especially in highspeed fluid flows. Shocks are often generated by supersonic or nearsonic 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 reflectiondiffraction 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 reflectiondiffraction 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 GuiQiang G. Chen and his collaborator Mikhail Feldman (University of WisconsinMadison) have introduced new ideas and developed techniques for solving fundamental open problems for multidimensional (MD) shock reflectiondiffraction and related free boundary problems for nonlinear conservation laws of mixed hyperbolicelliptic 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 reflectiondiffraction by wedges and employed the approach to solve rigorously the problem with largeangle 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 reflectiondiffraction, 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.
