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.

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?

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.