## When does one of the central ideas in economic theory work?

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?

## Fano Manifolds Old and New

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

## Tensor clustering of breast cancer data for network construction - Oxford Mathematics Research

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.

## Sharp rates of energy decay for damped waves - Oxford Mathematics Research

Differential equations arising in physics and elsewhere often describe the evolution in time of quantities which also depend on other (typically spatial) variables. Well known examples of such *evolution equations* include the heat equation and the wave equation.

## Classification of geometric spaces in F-theory

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

## Reconstructing the number of edges from a partial deck

Oxford Mathematician Carla Groenland talks about her and Oxford colleagues' work on graph reconstruction.

A graph $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$ which may connect two (distinct) vertices. (There are no self-loops or multiple edges.)

## Constraining Nonequilibrium Physics

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.

## Simulating polarized light

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 Geometry of Differential Equations - Oxford Mathematics Research by Paul Tod

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.

## D-modules on rigid analytic spaces - where algebra and geometry meet number theory

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

## What happens when aircraft fly through clouds? From high speed drop impact to icing prevention

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.

## Random minimum spanning trees

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.

## Understanding the complicated behaviour of free liquid sheets

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.

## Using smartphones for detecting the symptoms of Parkinson’s disease

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

## Noisy brains - Fast white noise generation for modelling uncertainty in the fluid dynamics of the brain

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.

## Connecting the Dots with Pick's Theorem

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.

## From knots to foliations - understanding the complexity of tangled ropes

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

## Teaching the machines - topology and signatures for parameter inference in dynamical systems

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.

## The Mathematics of Smoothies - the Dynamics of Particle Chopping in Blenders and Food Processors

Have you ever forgotten to replace the lid of the blender before beginning to puree your mango and passion-fruit smoothie? If you have, you'll have witnessed the catastrophic explosion of fruit and yoghurt flung haphazardly around the kitchen in an unpredictable manner. This is a consequence of the complicated and turbulent fluid dynamics present within the machine, the exact behaviour of which is unknown.

## What is Representation Theory and how is it used? Oxford Mathematics Research investigates

Oxford Mathematician Karin Erdmann specializes in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology). Here she discusses her latest work.

## Using mathematical modelling to identify future diagnoses of Alzheimer's disease

Oxford Mathematician Paul Moore talks about his application of mathematical tools to identify who will be affected with Alzheimer's.

## The twist and turns of curved objects - Oxford Mathematics research investigates the stability and robustness of everted spherical caps

Everyday life tells us that curved objects may have two stable states: a contact lens (or the spherical cap obtained by cutting a tennis ball, see picture) can be turned ‘inside out’. Heuristically, this is because the act of turning the object inside out keeps the central line of the object the same length (the centreline does not stretch significantly). Such deformations are called ‘isometries’ and the ‘turning inside out’ (or everted) isometry of a thin shell is often referred to as mirror buckling.

## Tricks of the Tour - optimizing the breakaway position in cycle races using mathematical modelling

Cycling science is a lucrative and competitive industry in which small advantages are often the difference between winning and losing. For example, the 2017 Tour de France was won by a margin of less than one minute for a total race time of more than 86 hours. Such incremental improvements in performance come from a wide range of specialists, including sports scientists, engineers, and dieticians. How can mathematics assist us?

## Do stochastic systems converge to a well-defined limit? Oxford Mathematics Research investigates

Oxford Mathematician Ilya Chevyrev talks about his research into using stochastic analysis to understand complex systems.

## Incorporating stress-assisted diffusion in cardiac models

Oxford Mathematician Ricardo Ruiz Baier, in collaboration mainly with the biomedical engineer Alessio Gizzi from Campus Bio-Medico, Rome, have come up with a new class of models that couple diffusion and mechanical stress and which are specifically tailored to the study of cardiac electromechanics.

## Knots and surfaces - the fascinating topology of n-manifolds

Oxford Mathematician Andras Juhasz discusses and illustrates his latest research into knot theory.

## The brains of the matter. Understanding the cerebral cortex

The brain is the most complicated organ of any animal, formed and sculpted over 500 million years of evolution. And the cerebral cortex is a critical component. This folded grey matter forms the outside of the brain, and is the seat of higher cognitive functions such as language, episodic memory and voluntary movement.

## Oxford Mathematics Research studies Unitary representations of Lie groups

Oxford Mathematician Dan Ciubotaru talks about his recent research in Representation Theory.

## Gauge theory - where particle physics and mathematics meet

Oxford Mathematician Yuuji Tanaka describes his part in the advances in our understanding of gauge theory.

## Assessing the impact of local planning on housing delivery and affordability

The investment decisions made by the construction sector have an obvious impact on the supply of housing. Furthermore, Local Planning Authorities play a fundamental role in shaping this supply via town planning and, in particular, by approving or rejecting planning applications submitted by developers. However, the role of these two factors, as well as their interaction, has so far been largely neglected in models of the housing market.

## Modelling outbreaks of infectious disease - diagnostic tests key for epidemic forecasting

Precise forecasting in the first few days of an infectious disease outbreak is challenging. However, Oxford Mathematical Biologist Robin Thompson and colleagues at Cambridge University have used mathematical modelling to show that for accurate epidemic prediction, it is necessary to develop and deploy diagnostic tests that can distinguish between hosts that are healthy and those that are infected but not yet showing symptoms.

## West Nile Virus and understanding the spread of infection

West Nile virus (WNV) is responsible for viral encephalitis in humans, a condition that causes inflammation of the brain and can have longer-lasting physical effects. WNV is also related to similar viruses such as Dengue and Zika that are also of significant public health concern.

## Oxford Mathematics Research looks at Ricci Flow

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Andrew Dancer discusses his work on Ricci Flow.

## A continuum of expanders

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician David Hume discusses his work on networks and expanders.

## How our immune systems could help us understand crime

Taxation and death may be inevitable but what about crime? It is ubiquitous and seems to have been around for as long as human beings themselves. A disease we cannot shake. However, therein lies an idea, one that Oxford Mathematician Soumya Banerjee and colleagues have used as the basis for understanding and quantifying crime.

## Knots and the nature of 3-dimensional space

It is an intriguing fact that the 3-dimensional world in which we live is, from a mathematical point of view, rather special. Dimension 3 is very different from dimension 4 and these both have very different theories from that of dimensions 5 and above. The study of space in dimensions 2, 3 and 4 is the field of low-dimensional topology, the research area of Oxford Mathematician Marc Lackenby.

## Numerical Analyst Nick Trefethen on the pleasures and significance of his subject

Oxford Mathematician Nick Trefethen was recently awarded the George Pólya Prize for Mathematical Exposition by the Society for Industrial and Applied Mathematics (SIAM) "for the exceptionally well-expressed accumulated insights found in his books, papers, essays, and talks." Here Nick refllects on the award, his approach to mathematics and the ever-expanding role of Numercial Analysis in the world.

## How do biomembranes form micro-structures in our cells?

The human body comprises an incredibly large number of cells. Estimates place the number somewhere in the region of 70 trillion, and that’s even before taking into account the microbes and bacteria that live in and around the body. Yet inside each cell, a myriad of complex processes occur to conceive and sustain these micro-organisms.

## Some advice for gamblers from Oxford Mathematics

We all know there is no guaranteed way of beating the bank in a casino or predicting the tossing of a coin. Well maybe. Perhaps a little more thought and a large dose of mathematics could help optimise our strategies.

## Iteration of Quadratic Polynomials Over Finite Fields - new research from Professor Roger Heath-Brown

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Eminent Oxford Mathematician and number theorist Roger Heath-Brown discusses his latest work.

"Since retiring last September I've had plenty of time for research. Here is something I've been looking into.

## The physics of the frog and the lily pad

A resting frog can deform the lily pad on which it sits. The weight of the frog applies a localised load to the lily pad (which is supported by the buoyancy of the liquid below), thus deforming the pad. Whether or not the frog knows it, the physical scenario of a floating elastic sheet subject to an applied load is present in a diverse range of situations spanning a spectrum of length scales. At global scales the gravitational loading of the lithosphere by mountain ranges and volcanic sea mounts involve much the same physical ingredients.

## Creating successful cities - how mathematical modelling can help

Oxford Mathematician Neave O’Clery recently moved to Oxford from the Center for International Development at Harvard University where she worked on the development of mathematical models to describe the processes behind industrial diversification and economic growth. Here she discusses how network science can help us understand the success of cities, and provide practical tools for policy-makers.

## Mathematics and Politics: The International Congresses of Mathematicians

The International Congresses of Mathematicians (ICMs) take place every four years at different locations around the globe, and are the largest regular gatherings of mathematicians from all nations. However, as much as the assembled mathematicians may like to pretend that these gatherings transcend politics, they have always been coloured by world events: the congresses prior to the Second World War saw friction between French and German mathematicians, for example, whilst Cold War political tensions likewise shaped the conduct of later congresses.

## Oxford Mathematics Research - Rates of convergence in the method of alternating projections

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician David Seifert discusses his and his collaborator Catalin Badea's work.

## Improving the performance of solar cells

Organometal halide perovskite (OMHP) is hardly a household name, but this new material is the source of much interest, not least for Oxford Applied Mathematicians Victor Burlakov and Alain Goriely as they model the fabrication and operation of solar cells.

## Well behaved cities - what all cities have in common

How are people, infrastructure and economic activity organised and interrelated? It is an intractable problem with ever-changing infinite factors of history, geography, economy and culture playing their part.

## Mitigating the impact of buy-to-let on the housing market

Much has been written about the buy-to-let sector and its role in encouraging both high levels of leverage and increases in house prices. Now Oxford Mathematician Doyne Farmer and colleagues from the Institute for New Economic Thinking at the Oxford Martin School and the Bank of England have modelled that impact.

## Did Value at Risk cause the crisis it was meant to avert?

## Functional calculus for operators

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. Th

## The Framed Standard Model for particles with possible implications for dark matter

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

## Knots and almost concordance

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.

## Improving techniques for optimising noisy functions

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.

## Structure or randomness in metric diophantine approximation?

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 $\pi \approx 3.14$.

## Categorification and Quantum Field Theories

Oxford Mathematician Elena Gal talks about her recently published research.

## Are there generic features of diseases such as Alzheimer's that can be explained from simple mechanistic models?

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.

## Flux-dependent graphs - enabling a better understanding of cellular metabolism

How does cellular metabolism change in different environments? Metabolism is the result of a highly enmeshed set of biochemical reactions, naturally amenable to graph-based analyses. Yet there are multiple ways to construct a graph representation from a metabolic model.

## How are trading strategies in electronic markets affected by latency?

Oxford Mathematicians Álvaro Cartea and Leandro Sánchez-Betancourt talk about their work on employing stochastic optimal control techniques to mitigate the effects of the time delay when receiving information in the marketplace and the time delay when sending instructions to buy or sell financial instruments on electronic exchanges.

## Mechanistic models versus machine learning: a fight worth fighting for the biological community?

90% of the world’s data have been generated in the last five years. A small fraction of these data is collected with the aim of validating specific hypotheses. These studies are led by the development of mechanistic models focussed on the causality of input-output relationships. However, the vast majority of the data are aimed at supporting statistical or correlation studies that bypass the need for causality and focus exclusively on prediction.

## Lifting the curse of ill-posedness by hybridization

At the beginning of the 20th century, Jacques Hadamard gave the definition of well-posed problems, with a view to classifying “correct” mathematical models of physical phenomena. Three criteria should be fulfilled: a solution exists, that solution is unique, and it should depend continuously on the parameters.

## The ‘shear’ brilliance of low head hydropower

The generation of electricity from elevated water sources has been the subject of much scientific research over the last century. Typically, in order to produce cost-effective energy, hydropower stations require large flow rates of water across large pressure drops. Although there are many low head sites around the UK, including numerous river weirs and potential tidal sites, the pursuit of low head hydropower is often avoided because it is uneconomic.

## The contact-free knot - Oxford Mathematics Research explains

Knots are widespread, universal physical structures, from shoelaces to Celtic decoration to the many variants familiar to sailors. They are often simple to construct and aesthetically appealing, yet remain topologically and mechanically quite complex.

Knots are also common in biopolymers such as DNA and proteins, with significant and often detrimental effects, and biological mechanisms also exist for 'unknotting'.

## Understanding plasma-liquid interactions

Oxford Mathematician John Allen, Professor Emeritus of Engineering Science, talks about his work on the electrohydrodynamic stability of a plasma-liquid interface. His collaborators are Joshua Holgate and Michael Coppins at Imperial College.

## Combinatorics - past, present and future

Oxford Mathematician Katherine Staden provides a fascinating snapshot of the field of combinatorics, and in particular extremal combinatorics, and the progress that she and her collaborators are making in answering one of its central questions posed by Paul Erdős over sixty years ago.

## Wrinkly Impact - award-winning film from Oxford Mathematics

Oxford Mathematicians Dominic Vella and Finn Box together with colleague Alfonso Castrejón-Pita from Engineering Science in Oxford and Maxime Inizan from MIT have won the annual video competition run by the UK Fluids Network. Here they describe their work and the film.

## The mathematics of security - Oxford Mathematics researchers and undergraduates expose security flaw

As recent breaches have demonstrated, security will be one of the major concerns of our digital futures. The collective intelligence of the mathematical community is critical to finding these flaws. A group of Oxford Mathematicians, both researchers and undergraduates, have done just that.

## How understanding Oscillator Networks could help unlock the secrets of brain diseases

Oxford Mathematician Christian Bick talks about his and colleagues' research into oscillator networks and how it could be valuable in understanding diseases such as Parkinson's.

## Modelling the production of silicon in furnaces

How can solar panels become cheaper? Part of the cost is in the production of silicon, which is manufactured in electrode-heated furnaces through a reaction between carbon and naturally occurring quartz rock. Making these furnaces more efficient could lead to a reduction in the financial cost of silicon and everything made from it, including computer chips, textiles, and solar panels. Greater efficiency also means reduced pollution.

## How to make sense of our digital conversations

For many years networks have been a fruitful source of study for mathematicians, one of the first notable examples of network analysis being Leonard Euler's study of paths on the Königsberg bridges. Since that time the field of graph theory and network science has developed greatly and the problems we want to model have also changed.

## Exploring Steiner Chains with Möbius Transformations

Oxford Mathematician Kristian Kiradjiev has been awarded the Institute of Mathematics and its Applications (IMA) Early Career Mathematicians Catherine Richards Prize 2017 for his article on 'Exploring Steiner Chains with Möbius Transformations.' Here he explains his work.

## Hair today, gone tomorrow. But have scientists found a new way to stimulate hair growth?

How does the skin develop follicles and eventually sprout hair? Research from a team including Oxford Mathematicians Ruth Baker and Linus Schumacher addresses this question using insights gleaned from organoids, 3D assemblies of cells possessing rudimentary skin structure and function, including the ability to grow hair.

## Oxford Mathematics Research: Nikolay Nikolov on his latest research into Sofic Groups

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Nikolay Nikolov discusses his research in to Sofic Groups.

## The mathematics of abnormal skull growth

Mathematics is delving in to ever-wider aspects of the physical world. Here Oxford Mathematician Alain Goriely describes how mathematicians and engineers are working with medics to better understand the workings of the human brain and in particular the issue of abnormal skull growth.

## The mathematics of glass sheets - how to make their thickness uniform

Oxford Mathematician Doireann O'Kiely was recently awarded the IMA's biennial Lighthill-Thwaites Prize for her work on the production of thin glass sheets. Here Doireann describes her work which was conducted in collaboration with Schott AG.

## Mathematical physicist James Sparks talks about his research into exact results in the AdS/CFT correspondence

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician James Sparks discusses his latest work.

"Two great successes of 20th century theoretical physics are Quantum Field Theory and General Relativity.

## Modelling the architecture of the brain

Oxford Mathematicians Tamsin Lee and Peter Grindrod discuss their latest research on the brain, part of our series focusing on the complexities and applications of mathematical research and modelling.

## Using mathematical modelling to improve our understanding of radiotherapy

New methods for localising radiation treatment of tumours depend on estimating the spatial distribution of oxygen in the tissue. Oxford Mathematicians hope to improve such estimates by predicting tumour oxygen distributions and radiotherapy response using high resolution images of real blood vessel networks.

## Understanding the risks banks pose to the financial system

Systemic risk, loosely defined, describes the risk that large parts of the financial system will collapse, leading to potentially far-reaching consequences both within and beyond the financial system. Such risks can materialize following shocks to relatively small parts of the financial system and then spread through various contagion channels. Assessing the systemic risk a bank poses to the system has thus become a central part of regulating its capital requirements.

## Modelling the impact of scientific collaboration

If nations are to grow, both economically and intellectually, they must foster scientific creativity. To do that they must create scientific environments that stimulate collaboration. This is especially true of developing countries as they seek to prosper in a global economy.

## Mathematics and health promotion - discussing diabetes on Twitter

Social media for health promotion is a fast-moving, complex environment, teeming with messages and interactions among a diversity of users. In order to better understand this landscape a team of mathematicians and medical anthropologists from Oxford, Imperial College and Sinnia led by Oxford Mathematician Mariano Beguerisse studied a collection of 2.5 million tweets that contain the term "diabetes".

## The magic of numbers - finding structure in randomness

Mathematics is full of challenges that remain unanswered. The field of Number Theory is home to some of the most intense and fascinating work. Two Oxford mathematicians, Ben Green and Tom Sanders, have recently made an important breakthrough in an especially tantalising problem relating to arithmetic structure within the whole numbers.

## Hummingbirds, umbrellas and hopper poppers do it. But why not as quickly as expected?

Many elastic structures have two possible equilibrium states. For example umbrellas that become inverted in a sudden gust of wind, nanoelectromechanical switches, origami patterns and even the hopper popper, which jumps after being turned inside-out. These systems typically move from one state to the other via a rapid ‘snap-through’. Snap-through allows plants to gradually store elastic energy, before releasing it suddenly to generate rapid motions, as in the Venus flytrap .

## The mathematics of species extinction

Correctly predicting extinction is critical to ecology. Claim extinction too late, and you may be taking resources away from a species that actually could be saved. Claim extinction too early, and you may cause the true extinction due to stopping resources, such as removing protection of its habitat.

## Scientists discover how a common garden weed expels its seeds at record speeds

Plants use many strategies to disperse their seeds, but among the most fascinating are exploding seed pods. Scientists had assumed that the energy to power these explosions was generated through the seed pods deforming as they dried out, but in the case of ‘popping cress’ (*Cardamine hirsuta*) this turns out not to be so. These seed pods don’t wait to dry before they explode.

## Are big-city transportation systems too complex for human minds?

Many of us know the feeling of standing in front of a subway map in a strange city, baffled by the multi-coloured web staring back at us and seemingly unable to plot a route from point A to point B. Now, a team of physicists and mathematicians has attempted to quantify this confusion and find out whether there is a point at which navigating a route through a complex urban transport system exceeds our cognitive limits.

## When a droplet hits a surface

## Comparing the social structure of different cities

People make a city. Each city is as unique as the combination of its inhabitants. Currently, cities are generally categorised by size, but research by Oxford Mathematicians Peter Grindrod and Tamsin Lee on the social networks of different cities shows that City A, which is twice the size of City B, may not necessarily be accurately represented as an amalgamation of two City Bs.

## Predicting the spread of brain tumours

Glioblastoma is an aggressive form of brain tumour, which is characterised by life expectancies of less than 2 years from diagnosis and currently has no cure. The only intervention available to a patient is having the infected area of their brain cut away as soon as the tumour cells are observed.

## Predicting and managing energy use in a low-carbon future

If effectively harnessed, increased uptake of renewable generation, and the electrification of heating and transport, will form the bedrock of a low carbon future. Unfortunately, these technologies may have undesirable consequences for the electricity networks supplying our homes and businesses. The possible plethora of low carbon technologies, like electric vehicles, heat pumps and photovoltaics, will lead to increased pressure on the local electricity networks from larger and less predictable demands.

## How weights and pulleys might explain the hunting techniques of toads

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. In their recently published paper Oxford Mathematicians Dominic Vella and Alain Goriely and colleagues looked at the dynamics of the chain when one of the weights is removed and thus one end is pulled with constant acceleration.

## From Birds to Bacteria: Modelling Migration at Many Scales

The use of mathematical models to describe the motion of a variety of biological organisms has been the subject of much research interest for several decades. If we are able to predict the future locations of bacteria, cells or animals, and then we subsequently observe differences between the predictions and the experiments, we would have grounds to suggest that the local environment has changed, either on a chemical or protein scale, or on a larger scale, e.g.

## The mathematics of violent plastic deformation

This picture shows the "Z" machine at Sandia Labs in New Mexico producing, for a tiny fraction of a second, 290 TW of power - about 100 times the average electricity consumption of the entire planet. This astonishing power is used to subject metal samples to enormous pressures up to 10 million atmospheres, causing them to undergo violent plastic deformation at velocities up to 10 km/s. How should such extreme behaviour be described mathematically?

## Multiple zeta values in deformation quantization

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...**

## Stochastic homogenization: Deterministic models of random environments

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 fluid mechanics of kidney stone removal

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.

## How to sweep up your fusion reaction

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.

## How to control chaos in your body

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.

## How do Nodal lines for eigenfunctions bring together so many facets of mathematics?

Oxford Mathematician Riccardo W. Maffucci is interested in `Nodal lines for eigenfunctions', a multidisciplinary topic in pure mathematics, with application to physics. Its study is at the interface of probability, number theory, analysis, and geometry. The applications to physics include the study of ocean waves, earthquakes, sound and other types of waves. Here he talks about his work.

## Tinkering with postulates. How some mathematics is now redundant. Or is it?

At the beginning of the twentieth century, some minor algebraic investigations grabbed the interest of a small group of American mathematicians. The problems they worked on had little impact at the time, but they may nevertheless have had a subtle effect on the way in which mathematics has been taught over the past century.

## Following up Turing - how reaction-diffusion models generate complex patterns

In a seminal 1952 paper, Alan Turing mathematically demonstrated that two reacting chemicals in a spatially uniform mixture could give rise to patterns due to molecular movement, or diffusion. This is a particularly striking result, as diffusion is considered to be a stabilizing mechanism, driving systems towards uniformity (think of a drop of dye spreading in water).

## How do node attributes mix in large-scale networks? Oxford Mathematics Research investigates

In this collaboration with researchers from the University of Louvain, Renaud Lambiotte from Oxford Mathematics explores the mixing of node attributes in large-scale networks.

## How do airlines gauge unknown demand?

Oxford Mathematicians Ilan Price and Jaroslav Fowkes discuss their work on unconstraining demand with Gaussian Processes.

"One of the key revenue management challenges which airlines, hotels, cruise ships (and other industries) all share is the need to make business decisions in the face of constrained (or censored) demand data.

## The Conformal Bootstrap - Oxford Mathematician Luis Fernando Alday explains

What does boiling water have in common with magnets and the horizon of black holes? They are all described by conformal field theories (CFTs)! We are used to physical systems that are invariant under translations and rotations. Imagine a system which is also invariant under scale transformations. Such a system is described by a conformal field theory. Remarkably, many physical systems admit such a description and conformal field theory is ubiquitous in our current theoretical understanding of nature.

## New Cryptosystems to ensure Data Security in the Post-Quantum World

Oxford Mathematician Ali El Kaafarani explains how mathematics is tackling the issue of post-quantum digital security.

"Quantum computers are on their way to us, not from a galaxy far far away; they are literally right across the road from us in the Physics Department of Oxford University.

## Oxford Mathematics Research probes the inner secrets of L-functions

Oxford Mathematician Tom Oliver talks about his research in to the rich mine of mathematical information that are L-functions.

## Why your morning cup of coffee sloshes

Americans drink an average of 3.1 cups of coffee per day (and mathematicans probably even more). When carrying a liquid, common sense says walk slowly and refrain from overfilling the container. But easier said than followed. Cue sloshing.

## What do fireflies and viruses have in common?

Oxford Mathematician Soumya Banerjee talks about his current work in progress.

"On warm summer days, fireflies mesmerise us with their glowing lights. They produce this cold light using a light-emitting molecule, the luciferin, and a complementary enzyme, luciferase. This process is known as bioluminescence.

## Developing particle-based software with Aboria

Over the last five decades, software and computation has grown to become integral to the scientific process, for both theory and experimentation. A recent survey of RCUK-funded research being undertaken in 15 Russell Group universities found that 92% of researchers used research software, 67% reported that it was fundamental to their research, and 56% said they developed their own software.

## Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials

Oxford Mathematician Dmitry Belyaev is interested in the interface between analysis and probability. Here he discusses his latest work.

"There are two areas of mathematics that clearly have nothing to do with each other: projective geometry and conformally invariant critical models of statistical physics. It turns out that the situation is not as simple as it looks and these two areas might be connected.

## Heterogeneity in cell populations - a cautionary tale

Researchers from Oxford Mathematics and Imperial College London have provided a “'mathematical thought experiment' to inspire caution in biologists measuring heterogeneity in cell populations.

## Exploding the myths of Ada Lovelace’s mathematics

Ada Lovelace (1815–1852) is celebrated as “the first programmer” for her remarkable 1843 paper which explained Charles Babbage’s designs for a mechanical computer. New research reinforces the view that she was a gifted, perceptive and knowledgeable mathematician.

## Shapes and Numbers - Oxford Mathematics Research considers number theory and topology

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics, we look at Oxford Mathematician Minyhong Kim's research in to the relationship between number theory and topology. Minhyong Kim is Professor of Number Theory here in Oxford and Fellow of Merton College.

It is probably well-known that number theory is the source of some of the oldest and most accessible questions in mathematics:

## How fast does the Greenland Ice Sheet move?

Governments around the world are seeking to address the economic and humanitarian consequences of climate change. One of the most graphic indications of warming temperatures is the melting of the large ice caps in Greenland and Antarctica. This is a litmus test for climate change, since ice loss may contribute more than a metre to sea-level rise over the next century, and the fresh water that is dumped into the ocean will most likely affect the ocean circulation that regulates our temperature.

## Can Big Data root out corruption in Africa?

Many anticorruption advocates are excited about the prospects that “big data” will help detect and deter graft and other forms of malfeasance. But good data alone isn’t enough. To be useful, there must be a group of interested and informed users, who have both the tools and the skills to analyse the data to uncover misconduct, and then lobby governments and donors to listen to and act on the findings.

## The mathematics of why our grandmothers love us

Michael Bonsall, Professor of Mathematical Biology at Oxford University's Department of Zoology, discusses his research in population biology, what it tells us about species evolution and, in particular, why grandmothering is important to humans. His research was done in conjunction with Oxford Mathematician Jared Field.

"What is mathematical biology?

## The mathematics of sperm control

From studying the rhythmic movements, researchers at the Universities of York, Birmingham, Oxford and Kyoto University, Japan, have developed a mathematical formula which makes it easier to understand and predict how sperm make the journey to fertilise an egg. This knowledge will help scientists to gauge why some sperm are successful in fertilisation and others are not.

## Why shells behave unexpectedly when poked - Oxford Mathematics Research

The classic picture of how spheres deform (e.g. when poked) is that they adopt something called 'mirror buckling' - this is a special deformation (an isometry) that is geometrically very elegant. This deformation is also very cheap (in terms of the elastic energy) and so it has long been assumed that this is what a physical shell (e.g. a ping pong ball or beach ball) will do when poked. However, experience shows that actually many shells don’t adopt this state - instead, beach balls wrinkle and ping pong balls crumple.

## Modelling the growth of blood vessels in tumours

Cancer is a complex and resilient set of diseases and the search for a cure requires a multi-strategic approach. Oxford Mathematicians Lucy Hutchinson, Eamonn Gaffney, Philip Maini and Helen Byrne and Jonathan Wagg and Alex Phipps from Roche have addressed this challenge by focusing on the mathematical modelling of blood vessel growth in cancer tumours.

## Applied mathematics: don’t think twice, it’s all right

In an interview with Rolling Stone Magazine in 1965, Bob Dylan was pushed to define himself: Do you think of yourself primarily as a singer or a poet? To which, Dylan famously replied: Oh, I think of myself more as a song and dance man, y’know. Dylan’s attitude to pigeonholing resonates with many applied mathematicians. I lack the coolness factor of Dylan, but if pushed about defining what kind of mathematician I am, I would say: Oh, I think myself more as an equation and matrix guy, y’know.

## Oxford Mathematics Research - On the null origin of the ambitwistor string

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Eduardo Casali discusses his work.

## The impact of mathematics – human interactions!

Think of a mathematician and you might imagine an isolated individual fueled by coffee whose immaculate if incomprehensible papers may, in the fullness of time, via a decades-long dry chain of citations, be made use of by an industrialist (via one or two other dedicated mathematicians).

## The Mathematics of Shock Reflection-Diffraction and von Neumann’s Conjectures

As part of our series of research articles deliberately focusing on the rigour and complexity of mathematics and its problems, Oxford Mathematician Gui-Qiang G Chen discusses his work on the Mathematics of Shock Reflection-Diffraction.

## Improving the quality and safety of x-rays

X-ray imaging is an important technique for a variety of applications including medical imaging, industrial inspection and airport security. An X-ray image shows a two-dimensional projection of a three-dimensional body. The original 3D information can be recovered if multiple images are given of the same object from different viewpoints. The process of recovering 3D information from a set of 2D X-ray projections is called Computed Tomography (CT).

## Mathematics enables a better understanding of damage during brain surgery

For over a hundred years, when confronted by swelling in the brain, surgeons more often than not have resorted to decompressive craniectomy, the traditional route to reducing swelling by removing a large part of the skull. However, while this might be the standard procedure, its failure rate has been worryingly high, primarily because the consequences on the rest of the brain have been poorly understood.

## Using geometry to choose the best mathematical model

Across the physical and biological sciences, mathematical models are formulated to capture experimental observations. Often, multiple models are developed to explore alternate hypotheses. It then becomes necessary to choose between different models.

## Unleashing the mathematics of the chameleon's tongue

The chameleon's tongue is said to unravel at the sort of speed that would see a car go from 0-60 mph in one hundredth of a second – and it can extend up to 2.5 body lengths when catching insects. Oxford Mathematicans Derek Moulton and Alain Goriely have built a mathematical model to explain its secrets.

## The mathematics of poaching and gamekeeping

How do we stop poaching? You may think the answer lies in finding a way of giving gamekeepers an advantage over poachers. Oxford Mathematician Tamsin Lee and David Roberts from the University of Kent decided to look at the interaction between rhino poachers and a gamekeeper to predict the outcome of the battle. Their conclusions suggest alternative ways of tackling the problem.

## The universal structure of language

Semantics is the study of meaning as expressed through language, and it provides indirect access to an underlying level of conceptual structure. However, to what degree this conceptual structure is universal or is due to cultural histories, or to the environment inhabited by a speech community, is still controversial. Meaning is notoriously difficult to measure, let alone parameterise, for quantitative comparative studies.

## How predictable is technological progress?

Everyone knows that Moore’s law says that computers get cheaper at an exponential rate. What is not as well known is that many other technologies that have nothing to do with computers obey a similar law. Costs for DNA sequencing, some forms of renewable energy, chemical processes and consumer goods have also dropped at an exponential rate, even if the rates vary and are typically slower than for computers.

## Mitigating the impact of frost heave

Frost heave is a common problem in any country where the temperature drops below 0 degrees Celsius. It’s most commonly known as the cause of potholes that form in roads during winter, costing billions of dollars worth of damage worldwide each year. However, despite this, it is still not well understood. For example, the commonly accepted explanation of how it occurs is that water expands as it freezes, and this expansion tears open the surrounding material.

## Constructing reaction systems - an inverse problem in mathematics

There is a wide class of problems in mathematics known as inverse problems. Rather than starting with a mathematical model and analysing its properties, mathematicians start with a set of properties and try to obtain mathematical models which display them. For example, in mathematical chemistry researchers try to construct chemical reaction systems that have certain predefined behaviours. From a mathematical point of view, this can be used to create simplified chemical systems that can be used as test problems for different mathematical fields.

## Mathematics enables faster computer simulations of biology

Numerous processes across both the physical and biological sciences are driven by diffusion, for example transport of proteins within living cells, and some drug delivery mechanisms. Diffusion is an unguided process which is of great importance at small spatial scales.

## Mathematical theories of consciousness

How a complex dynamic network such as the human brain gives rise to consciousness has yet to be established by science. A popular view among many neuroscientists is that, through a variety of learning paradigms, the brain builds relationships and in the context of these relationships a brain state acquires meaning in the form of the relational content of the corresponding experience.

## How do networks shape the spread of disease and gossip?

A new approach to exploring the spread of contagious diseases or the latest celebrity gossip has been tested using London’s street and underground networks. Results from the new approach could help to predict when a contagion will spread through space as a simple wave (as in the Black Death) and when long-range connections, such as air travel, enable it to seemingly jump over long distances and emerge in locations far from an initial outbreak.