Electrochemical systems are essential to modern life. They encompass energy technologies like batteries and fuel cells as well as systems for desalination, mineral extraction, and chemical sensors. The fundamental physics of these systems relies on two components: electrodes and an electrolyte. You might be familiar with electrolytes in the context of sports drinks. Fundamentally, a liquid electrolyte, whether in a drink or an electrochemical device, is an ionic salt solution like dissolved sodium chloride.
The mathematician Oswald Veblen (1880–1960) is not widely remembered today, except perhaps by those who study geometrical or topological topics that relate to his work. In his day, however, Veblen was a major figure in American mathematics.
In this case study Jennifer Pi studies the complexity of the "classification programme" for certain objects called C$^*$-algebras.
Model merging asks whether specialised models can be combined into a single model that performs well across tasks, without retraining from scratch.
Although there are multiple ways of defining a notion of complexity on matrices, there is one which is widely viewed among mathematicians as being the canonical notion: the rank from linear algebra.
In our latest research case study, Harini introduces the concept of isomonodromic deformations and explain the role they play in the study of random curves.
Quantum field theories are full of mathematical riches, so long as one is clever and knows where to look. In this case study, I describe recent work inspired by the physics of four-dimensional superconformal field theory that uncovers an appearance of some of the formal structures coming from Kähler geometry within the theory of vertex operator algebras.
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.