Brownian dynamics have become ubiquitous in the mathematical modelling of noisy real-world systems. Typically, one considers the interplay between the governing forces of the system and the random fluctuations that occur due to noise. This culminates in the mathematical framework of stochastic differential equations (SDEs), which have found applications in finance, biology, and far beyond.
In this case study Chenjing explains the component lattice which involves a blend of Lie theory and stack theory.
Partial differential equations (PDEs), often regarded as the language of physics and engineering, encode how quantities such as velocity, temperature, pressure, or concentration evolve in space and time. PDEs provide the mathematical framework through which we model the real world. Yet, even when the governing equations are known, predicting their behaviour can be challenging. Konstantin Riedl investigates.
Quantum computers achieve a remarkable exponential speedup in integer factorisation (potentially making widely deployed cryptographic schemes vulnerable). Beyond that large-scale applications remain comparatively scarce, and if a fully error-corrected quantum computer were available it is not clear what 'killer app' it would be used for.
Many systems that govern crucial aspects of our lives can be seen as interacting dynamical units. This includes systems on a vast variety of scales, from billions of tiny interacting neural cells that are a critical part of our brain to the large scale communication networks that keep our world running. While we may understand the behaviour of each dynamical unit, it is crucial to understand the emergent collective dynamics of all units together.
Einstein's equations describe how the curvature of spacetime relates to energy and mass. However, as analytical solutions are rarely feasible, numerical solutions of these equations on computers are indispensable and are crucial in modern astrophysics, such as the detection of gravitational waves. But there are problems...
I study the large scale geometry of infinite groups and spaces, focusing on quasi-isometries, which are maps between groups or spaces that preserve the large scale geometry. Since quasi-isometries need not be continuous, distinguishing groups up to quasi-isometries can be challenging. This motivates considering invariants, that is, properties preserved under quasi-isometries.
I work in the field of geometric group theory. This is a pretty broad heading, but for me it means that the goal is to understand infinite groups, and the strategy is to get them to act on nice metric spaces that we know a lot about.
Historically, buckling has been regarded as a route to failure and structural collapse, and its study has been driven by the need to prevent its catastrophic consequences. However, in recent years, a novel perspective has emerged. The control of buckling, and mechanical instabilities more broadly, has potential to actuate new functionalities in engineered structures. This paradigm shift has also led to the identification of buckling as a means to render functionality in natural systems.
Cycling is a sport where victory often hinges on marginal gains. In elite races like the Tour de France, while power output and aerodynamics are well-known performance factors another crucial, but less visible, element is risk. A crash, even if minor, can end a rider’s race. Can mathematics help optimise racing strategies in a world where both energy and safety must be balanced?
A new study uncovers a surprising connection between sliding droplets and migrating groups of cells. By combining live 3D imaging with mathematical modelling, scientists have revealed how cell collectives organise themselves in space and time, which depends on the cell collectives behaving like fluids.
Why do some memories last a lifetime while others fade away? A groundbreaking new study sheds light on this mystery by uncovering hidden patterns of brain activity that support long-term memory. Using a framework inspired by thermodynamics, scientists have developed a novel approach to understanding how different brain regions work together to shape cognition.
In this case study we survey the historical development of $\mathrm{Lip}(\gamma)$ functions, beginning with the work of Hassler Whitney from the 1930s and ending with some of the recent properties established by Terry Lyons and Andrew McLeod that are particul
A team of Oxford Mathematicians together with colleagues from the Oxford Botanic Garden and the University of Manchester has solved a mystery that has intrigued scientists for centuries. The squirting cucumber (Ecballium elaterium, from the Greek ‘ekballein,’ meaning to throw out) is named for the ballistic method the species uses to disperse its seeds. But how does it do it?
In a new study, Oxford Mathematician Coralia Cartis and Samar Khatiwala from Oxford's Department of Earth Sciences, together with colleagues and support from across the UK, Europe and the USA, have developed a novel approach to speed up the optimisation of ocean biogeochemical models - critical tools for predicting the impacts of climate change on marine ecosystems and the global carbon cycle.
Symmetry has been a guiding principle for physics ever since its modern conception in the work of Issac Newton, which eventually led to the reformulation of classical mechanics by Lagrange and Hamilton.
Mathematics and the natural world don't always see eye to eye; or rather shape to shape. When mathematicians study how tiles fit together, they keep it simple. Triangles, squares, and hexagons along with cubes and other polyhedra are built with sharp corners and flat faces. That works for maths.
But it doesn't work for Nature.
My recent research has been focused on the "sum-product phenomenon" which is a vast collection of results analysing the incongruence between additive and multiplicative structure over finite sets of numbers.