Inspired by jumping insects, Oxford Mathematicians have helped develop a miniature robot capable of leaping more than 40 times its body length - equivalent to a human jumping up to the 20th floor of a building. The innovation could be a major step forward in developing miniature robots for a wide range of applications.
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In 1900, David Hilbert posed his list of 23 mathematical problems. While some of them have been resolved in subsequent years, a few of his challenging questions remain unanswered to this day. One of them is Hilbert's 16th problem, which asks questions about the number of limit cycles that a system of ordinary differential equations can have. Such equations appear in modelling real-world systems in biological, chemical or physical applications, where their solutions are functions of time.
What do quantum mechanics, genetics, economic choice theory and ant behaviour have in common? Naturally, the first answer is that they have nothing at all to do with each other! Nonetheless, finding bridges between different fields is the bread and butter of the field of complex systems, an offshoot of statistical physics and other fields.
Algebraic topology is the study of the continuous shape of spaces by the discrete means of algebra. The beginning of modern algebraic topology can be traced back to an insight of Pontryagin in the 1930s which relates the global smooth geometry of manifolds to algebraic invariants associated to the local symmetries of those manifolds - this relation converts something smooth and geometric (called a manifold, potentially endowed with further structure) to something algebraic that can be written down with symbols and formulas, i.e.
Ecology, by my definition today, is the study of the phenomena, relationships, patterns and processes which constitute this living and breathing world which we all call home. Its reach is boundless, and the angles through which to look at ecological processes are diverse and span all academic disciplines. It is emerging as the central focus of the present day.
One of the current challenges in computational neuroscience is to understand how a huge number of interacting neurons can generate and sustain complex patterned activities that play important roles in the functioning of the brain.
Recently, two Oxford Mathematics postdoctoral research associates Johannes Borgqvist and Sam Palmer published an article entitled “Occam's razor gets a new edge: the use of symmetries in model selection” in the Journal of the Royal Society Interface.
Porous materials occur throughout nature and industry, from volcanic rocks, to sponges, through to filters; they comprise a solid structure, which could be like honeycomb or a packing of grains, with gaps (pores) throughout. These pores allows fluid and small contaminants carried by the fluid, such as salt in seawater, to travel through the material. The shape and size of these pores affects the passage of fluid, which in turn affects the path the contaminants take and how much becomes trapped within the solid structure.
Next time you lose your paddle whilst canoeing - don't despair. There is another way to push your canoe forwards: by jumping on it! Gunwale bobbing refers to the act of standing on the gunwales (side walls) of a canoe and forcing it into oscillations with one's legs. When forced at the right frequency, the canoe can surf from crest to crest of the generated wave field by pushing into positive surface gradients.
As cancer treatment improves and survival rates increase, the long- and short-term side-effects of these treatments become more of a concern. One such side-effect is autoimmune myocarditis, or cardiac muscle inflammation, which can occur in patients undergoing treatment with immune checkpoint inhibitors (ICIs), a class of drugs used in cancer therapy. Although autoimmune myocarditis is a rare side-effect affecting only 0.1-1% of patients being treated with ICIs, it has a high fatality rate at 25-50% of cases.
Topological data analysis enables deep quantification of vascular responses to drug and radiation treatment in 3D tumours. Oxford Mathematician Bernadette Stolz explains.
Many complex systems, such as cities, social media, animal social groups, or brains can be represented using the framework of network science, i.e., as a set of individual units linked together through certain types of interactions.
Quantum field theory (QFT) is a natural language for describing quantum physics that obeys special relativity. A modern perspective on QFT is provided by the renormalization group (RG) flow, which is a path defined on the coupling constant space and evolves from the ultraviolet (UV) to the infrared (IR) fixed point. In particular, the theories on the IR fixed point are scale-invariant and most of them are known to be promoted to a conformal field theory (CFT).
The study of finitely generated groups usually proceeds in two steps. Firstly, a class of spaces with some intrinsic geometric property is defined and understood, for example hyperbolic spaces or CAT(0) spaces. Secondly, we try to relate the geometry of the space to algebraic properties of groups acting properly discontinuously cocompactly (i.e. geometrically) on the space. For example, this gives rise to the well studied classes of hyperbolic groups and CAT(0) groups.
The virus causing the COVID-19 pandemic, SARS-CoV-2, is transmitted through virus-carrying respiratory droplets, which are released when an infected person coughs, sneezes, talks or breathes. Most of these droplets will fall to the ground within two metres, hence the guidelines to maintain social distancing. However, some droplets are small enough to float in the air. These droplets may remain airborne for hours and be carried throughout a room, leading to airborne transmission.
The British computer scientist Christopher Strachey (1916–1975) is frequently cited as having called for an end to an “artificial and injurious” separation of practical and theoretical work in programming.
The first months of 2020 brought the world to an almost complete standstill due to the occurrence and outbreak of the SARS-CoV-2 coronavirus, which causes the highly contagious COVID-19 disease. Despite the hopes that rapidly developing medical sciences would quickly find an effective remedy, the last two years have made it quite clear that, despite vaccines, this is not very likely.
Much of theoretical physics is built on the concept of perturbation theory. Essentially perturbation theory is the idea that we can find a small quantity in the physical system we wish to describe and use that to expand the equations describing the system. In this way we obtain a simpler set of equations to solve.
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.
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.