Define the prime-counting function $\pi(x)$ to be the number of natural primes less than or equal to $x$.
If you have ever watched the movie Transformers you may not have worried too much about whether it is realistic. Nevertheless, scientists have tried, and are still trying, to create objects that can transform their shape.
The Riemann zeta-function is defined by \[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}, \quad \text{for $\mathrm{Re}(s) > 1$}, \] and the domain is extended to all of $\mathbb{C}$ by analytic continuation. The zeros of the Riemann zeta-function are the subject of the Riemann hypothesis.
Industrial inkjets use inkjet technology - familiar to any office printer - to print or deposit materials as part of the manufacturing process of a product on a production line. It requires precise control of the formation and jetting of small droplets of liquid ink.
Curvature is a way of measuring the distortion of a space from being flat, and it is ubiquitous in Science. Ricci Curvature, in particular, appears in Einstein’s equations of General Relativity. It controls the heat diffusion in general ambient spaces and it plays a fundamental role in Hamilton-Perelman’s solution of the Poincaré conjecture and of Thurston’s geometrisation conjecture.
Today, two Oxford mathematicians András Juhász and Marc Lackenby published a paper in Nature in collaboration with authors from DeepMind and with Geordie Williamson at the University of Sydney.
Silicon is produced industrially in a submerged arc furnace (illustrated in Figure 1, left), with the heat required for the endothermic chemical reaction provided by an electric current. The high temperatures within the furnace (up to around 2000 K) prohibit observation of the internal conditions, so that mathematical modelling is a valuable tool to understand the furnace processes.
The 7m length by 2m diameter cylindrical Søderberg electrode is the secret behind a yearly production capacity of 215,000 tonnes of silicon for Elkem ASA, the third largest silicon producer in the world. This electrode operates continuously thanks to its raw material: carbon paste, whose viscosity depends very sensitively on the temperature.
A collaboration between pure and applied mathematicians, pathologists and clinicians, has shown how sophisticated techniques from computational algebraic topology can be applied to understand the spatial distributions of immune cell subtypes in the tumour microenvironment.
Alois Alzheimer called Alzheimer's disease (AD) the disease of forgetfulness in a 1906 lecture that would later mark its discovery. Alzheimer noticed the presence of aggregated protein plaques, made up of misfolded variants of amyloid-beta (A$\beta$) and tau ($\tau$P) proteins, in the brain of one of his patients. These plaques are thought to be the drivers of the overall cognitive decline that is observed in AD. AD is now one of the leading causes of death in many developed countries, including the United Kingdom.
Towards the end of the eighteenth century, French mathematician and engineer Gaspard Monge considered a problem. If you have a lot of rubble, you would like to have a fort, and you do not like carrying rocks very far, how do you best rearrange your disorganised materials into organised walls? Over the two centuries since then, his work has been developed into the rich mathematical theory of optimal transport.
In quantum many body physics, we look for universal features that allow us to classify complex quantum systems. This classification leads to phase diagrams of quantum systems. These are analogous to the familiar phase diagram of water at different temperatures and pressures, with ice and vapour constituting two phases. Quantum phase diagrams correspond to the different phases of matter at zero temperature, where the system is in its lowest energy state (usually called the ground state).
One of the main themes of geometry in recent years has been the appearance of unexpected dualities between different geometric spaces arising from ideas in mathematical physics. One famous such example is mirror symmetry. Another kind of duality, which I am currently investigating with collaborators from Oxford and Imperial College, is symplectic duality.
Knot theory studies embeddings of the circle into the three dimensional space and the first knot invariant was the Alexander polynomial. The world of quantum invariants started with the milestone discovery of the Jones polynomial and was expanded by Reshetikhin and Turaev’s algebraic construction which starts from a quantum group and leads to link invariants.
Oxford Mathematician Lukas Brantner explains how generalised Lie algebras lead to new insights in Galois theory, deformation theory, and the theory of configuration spaces. Lukas has just been awarded a Royal Society University Research Fellowship.
Oxford Mathematician Aymeric Vie, first year DPhil student at the Centre for Doctoral Training, Mathematics of Random Systems, describes his work on the population network structure of genetic algorithms.This work identifies new ways to improve the performance of those stochastic algorithms, and has received a Best Paper Award at the Genetic and Evolutionary Computation Conference 2021.
Oxford Mathematician Connor Behan discusses the ways in which a free quantum field can be coupled to a spatial boundary. His recent work with Lorenzo di Pietro, Edoardo Lauria and Balt van Rees sheds light on this question using the non-perturbative bootstrap technique.
What takes a mathematician to the Arctic? In short, context. The ice of the Arctic Ocean has been a rich source of mathematical problems since the late 19$^{th}$ century, when Josef Stefan, aided by data from expeditions that went in search of the Northwest Passage, developed the classical Stefan problem. This describes the evolution of a moving boundary at which a material undergoes a phase change. In recent years, interest in the Arctic has only increased, due to the rapid changes occurring there due to climate change.