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. This is t

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.

Homogenization theory aims to understand the properties of materials with complicated microstructures, such as those arising from flaws in a manufacturing process or from randomly deposited impurities. The goal is to identify an effective model that provides an accurate approximation of the original material. Oxford Mathematician Benjamin Fehrman discusses his research.

"The practical considerations for identifying a simplified model are twofold:

The discomfort experienced when a kidney stone passes through the ureter is often compared to the pain of childbirth. Severe pain can indicate that the stone is too large to naturally dislodge, and surgical intervention may be required. A ureteroscope is inserted into the ureter (passing first through the urethra and the bladder) in a procedure called ureteroscopy. Via a miniscule light and a camera on the scope tip, the patient’s ureter and kidney are viewed by a urologist.

The Sun has been emitting light and illuminating the Earth for more than four billion years. By analyzing the properties of solar light we can infer a wealth of information about what happens on the Sun. A particularly fascinating (and often overlooked) property of light is its polarization state, which characterizes the orientation of the oscillation in a transverse wave. By measuring light polarization, we can gather precious information about the physical conditions of the solar atmosphere and the magnetic fields present therein.

The problem of optimisation – that is, finding the maximum or minimum of an ‘objective’ function – is one of the most important problems in computational mathematics. Optimisation problems are ubiquitous: traders might optimise their portfolio to maximise (expected) revenue, engineers may optimise the design of a product to maximise efficiency, data scientists minimise the prediction error of machine learning models, and scientists may want to estimate parameters using experimental data.

Diophantine approximation is about how well real numbers can be approximated by rationals. Say I give you a real number $\alpha$, and I ask you to approximate it by a rational number $a/q$, where $q$ is not too large. A naive strategy would be to first choose $q$ arbitrarily, and to then choose the nearest integer $a$ to $q \alpha$. This would give $| \alpha - a/q| \le 1/(2q)$, and $\pi \approx 3.14$.

If you type *fundamental anagram of calculus* into Google you will be led eventually to the string of symbols** 6accdæ13eff7i3l9n4o4qrr4s8t12ux**, probably accompanied by an explanation more or less as follows: this is a recipe for an *anagram* - take six copies of **a**, two of **c**, one of **d**, one of **æ** and so on, then rearrange these letters into a chunk of Latin.

Fusion energy may hold the key to a sustainable future of electricity production. However some technical stumbling blocks remain to be overcome. One central challenge of the fusion enterprise is how to effectively withstand the high heat load emanating from the core plasma. Even the sturdiest solid solutions suffer damage over time, which could be avoided by adding a thin liquid coating.

Oxford Mathematician Andreas Bode talks about his work in representation theory and its lesson for the interconnectness of mathematics.

Oxford Mathematician Elena Gal talks about her recently published research.

As you settle into your seat for a flight to a holiday destination or as part of yet another business trip, it is very easy to become absorbed by the glossy magazines or the novel you've been waiting forever to start reading. Understandably,* *the phrase* "safety features on board this aircraft" *triggers a rather unenthusiastic response. But you may be surprised by some of the incredible technology just a few feet away that is there to make sure everything goes smoothly.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies.

Christina Goldschmidt from the Department of Statistics in Oxford talks about her joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (Paris Sorbonne University) and Gregory Miermont (ENS Lyon) on random minimum spanning trees.

Oxford Mathematician Siddharth Arora talks about his and his colleagues' research in to using smartphone technology to anticipate the symptoms of Parkinson’s disease.

Free suspended liquid films or sheets are often formed during industrial production of sprays as well as in natural processes such as sea spray. Early experimental and theoretical investigations of them were done by French physicist Felix Savart, who observed liquid sheets forming by a jet impact on a solid surface, or by two jets impacting each other (1833), and British physicist Arthur Mason Worthington, a pioneer in investigation of the crown splash forming after impact of a drop onto a liquid surface.

Over the last few years, the study of the physiological mechanisms governing the movement of fluids in the brain (referred to as the brain waterscape) has gained prominence. The reason? Anomalies in the brain fluid dynamics are related to diseases such as Alzheimer's disease, other forms of dementia and hydrocephalus. Understanding how the brain waterscape works can help discover how these diseases develop. Unfortunately, experimenting with the human brain in vivo is extremely difficult and the subject is still poorly understood.

Oxford Mathematician Kristian Kiradjiev has won the Graham Hoare Prize (awarded by the Institute of Mathematics and its Applications) for his article "Connecting the Dots with Pick's Theorem". The Graham Hoare Prize is awarded annually to Early Career Mathematicians for a brilliant Mathematics Today article. Kristian also won the award in 2017. Here he talks about his work.

Oxford Mathematician Mehdi Yazdi talks about his study of tangled ropes in 3-dimensional space.

In many natural systems, such as the climate, the flow of fluids, but also in the motion of certain celestial objects, we observe complicated, irregular, seemingly random behaviours. These are often created by simple deterministic rules, and not by some vast complexity of the system or its inherent randomness. A typical feature of such chaotic systems is the high sensitivity of trajectories to the initial condition, which is also known in popular culture as the butterfly effect.

Oxford Mathematician Vidit Nanda talks about his and colleagues Harald Oberhauser and Ilya Chevyrev's recent work combining algebraic topology and stochastic analysis for statistical inference from complex nonlinear datasets.

"It is not difficult to generate very complicated dynamics via very simple equations. Consider, for each parameter r > 0 and natural number n, the update rules