Oxford Mathematician Joni Teräväinen talks about his work concerning prime factorisations of consecutive integers and its applications in number theory.

Oxford Mathematician Ebrahim Patel talks about his work using max-plus algebra to improve airport scheduling and to define optimal railway networks.

In this collaboration with researchers from the Umeå University and the University of Zurich, Renaud Lambiotte from Oxford Mathematics explores the use of higher-order networks to analyse complex data.

Oxford Mathematician Benjamin Walker talks about his work on the automatic identification of flagella from images, opening up a world of data-driven analysis.

Oxford Mathematician Vinayak Abrol talks about his and colleagues's work on using mathematical tools to provide insights in to deep learning.

**Why it Matters!**

Oxford Mathematician Ric Wade explains how right-angled Artin groups, once neglected, are now central figures in low-dimension topology.

We’re all familiar with liquid droplets moving under gravity (especially if you live somewhere as rainy as Oxford). However, emerging applications such as lab-on-a-chip technologies require precise control of extremely small droplets; on these scales, the forces associated with surface tension become dominant over gravity, and it is therefore not practical to rely on the weight of the drops for motion.

The concept of equilibrium is one of the most central ideas in economics. It is one of the core assumptions in the vast majority of economic models, including models used by policymakers on issues ranging from monetary policy to climate change, trade policy and the minimum wage. But is it a good assumption?

Oxford Mathematician Thomas Prince talks about his work on the construction of Fano manifolds in dimension four and their connection with Calabi-Yau geometry.

Oxford Mathematicians Anna Seigal, Heather Harrington, Mariano Beguerisse Diaz and colleagues talk about their work on trying to find cancer cell lines with similar responses by clustering them with structural constraints.

Oxford Mathematician Yinan Wang talks about his and colleagues' work on classification of elliptic Calabi-Yau manifolds and geometric solutions of F-theory.

Oxford Mathematician Erik Panzer talks about his and colleagues' work on devising an algorithm to compute Kontsevich's star-product formula explicitly, solving a problem open for more than 20 years.

"The transition from classical mechanics to quantum mechanics is marked by the introduction of non-commutativity. For example, let us consider the case of a particle moving on the real line.

**From commutative classical mechanics...**

When mathematicians solve a differential equation, they are usually converting unbounded operators (such as differentiation) which are represented in the equation into bounded operators (such as integration) which represent the solutions. It is rarely possible to give a solution explicitly, but general theory can often show whether a solution exists, whether it is unique, and what properties it has. For this, one often needs to apply suitable (bounded) functions $f$ to unbounded operators $A$ and obtain bounded operators $f(A)$ with good properties.&

Oxford Mathematician Tsou Sheung Tsun talks about her work on building the Framed Standard Model and the exciting directions it has taken her.

Statistical mechanics (or thermodynamics) is a way of understanding large systems of interacting objects, such as particles in fluids and gases, chemicals in solution, or people meandering through a crowded street. Large macroscopic systems require prohibitively large systems of equations, and so equilibrium thermodynamics gives us a way to average out all of these details and understand the typical behaviour of the large scale system.

Knots are isotopy classes of smooth embeddings of $S^1$ in to $S^3$. Intuitively a knot can be thought of as an elastic closed curve in space, that can be deformed without tearing. Oxford Mathematician Daniele Celoria explains.

"Knots are ubiquitous in the study of the topological and geometrical properties of manifolds with dimension $3$ and $4$. This is due to the fact that they can be used to prescribe the attachment instructions for the "building blocks" of these spaces, through a process known as surgery.