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Mathematics Today

Mathematics Today is the membership publication of the Institute of Mathematics and its Applications.

Issued six times a year, this general interest mathematics publication provides articles, reports, reviews and news for mathematicians.

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Content from the current issue


While on holiday in Italy three years ago, my wife and I visited a magnificent exhibition of arts and crafts by 20th century Spanish painter, sculptor and designer, Salvador Dalí. Alongside his bizarre and incredibly expensive work, I was pleased to spot sculptures of the five regular polyhedra (Figure 1) that demonstrate his interest and ability in mathematics. Dalí also famously depicted a partial dodecahedron in his painting The Sacrament of the Last Supper and tackled other mathematical concepts including spheres, spirals, double helices, symmetries, catastrophe theory and the net (unfolded form) of a tesseract (four-dimensional cube).

2016 Nobel Prize in Physics

The 2016 Nobel Prize in Physics was awarded to Duncan Haldane, Michael Kosterlitz and David Thouless, three British theoretical physicists working in the US. Their citation reads ‘for theoretical discoveries of topological phase transitions and topological phases of matter’

Haldane, Kosterlitz and Thouless are condensed matter physicists. This is the study of ‘stuff’, of solids and liquids, and how their wonderfully diverse properties can emerge from the simple laws that govern the underlying atoms and electrons.

Can One Shape the Sound of a Drum?

Just 50 years ago, the Polish mathematician Kac [1] posed the intriguing question, ‘Can one hear the shape of a drum?’ It stimulated a variety of demonstrations that drums with differing irregular polygonal boundaries could have identical frequency spectra [2]. So, too, could circular drums, if only they could be made with inhomogeneous membranes having differing spatially-varying density or thickness profiles [3]. The answer to Kac was therefore an emphatic ‘No!’

This article derives from two related thoughts: first, that plastic membranes, now widely used as an alternative to traditional materials such as calfskin, could be made with prescribed thickness profiles by 3D printing and, secondly, that these profiles could shape the frequency spectrum.

Playing Tennis without Envy

A group of friends organises their tennis games by each submitting their availability over the week. They want to obtain an assignment such that: each game must be a doubles tennis match, i.e. requires four people, and nobody plays on a day he is unavailable. Can we construct assignments that will always produce efficient, fair, and envy-free outcomes? The answer is no, and extends to any sport that requires any group size.

In the June 2016 edition of the magazine Mathematics Today, Maher [1] described an algorithm to assign tennis doubles matches among his circle of friends. The algorithm takes as input the players’ availability for the week, and maximises the number of tennis games, subject to three constraints: 1) no agent plays more than once per day, 2) each match has exactly four players, and 3) no agent plays on a day he is not available.

Full contents page of the December 2016 printed issue
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Content from the October issue


Transportation is often described as an umbrella discipline because transport professionals come from many and varied backgrounds. Within academia transport groups are often located within either civil engineering or geography departments due to early work in road building and town planning and more recently within business schools due to the need for economics in transport. There are also transport psychologists, sociologists, environmentalists etc., and of course mathematics and mathematical modelling permeate throughout all fields within transportation studies.

An Interview with Maria Bruna

Maria Bruna is the first mathematician to win a prestigious L’Oréal-UNESCO UK and Ireland for Women in Science Fellowship. The Fellowships are designed to provide flexible financial help to outstanding female postdoctoral scientists to continue research in their chosen fields.

Dr Maria Bruna is a Junior Research Fellow at St John’s College and an Affiliate Researcher at the Mathematical Institute, University of Oxford. Maria tells us about the path she took to get there, her research and winning the L’Oréal-UNESCO Fellowship.

Congratulations on being the first mathematician awarded a L’Oréal-UNESCO Fellowship. What do you plan to do with it?

Thank you! The fellowship will allow me to establish a new research line with collaborators from the UK, Austria and Germany. I will use the fellowship money to organise a workshop in Oxford, fund research trips and cover part of the childcare costs of my 7-month old baby.

Modelling Road Accidents

Road accidents are, fortunately, rare events. They occur randomly but on a systematic background in space and time: that is, they generally occur more frequently at locations, and at times, when there is most traffic. If you were to produce a scatterplot of accident locations in any region of the country you would broadly observe that the main road network was traced out, with clusters of accidents along busy stretches of road and around busy intersections. The annual numbers of collisions on Britain’s roads has reduced by 43% over the period from 1979 to 2015, as may be seen in Figure 1, and fatal and serious collisions by around 70%, even though vehicle mileage has been increasing by on average around 1% per year.

Network Models of Route Choice

Network models are used in transport studies to explore the effects of individual travellers’ route choice. These model the likely consequences of changes in the demand for travel, facilities provided, and ways in which demand is assigned. This enables planners to anticipate the response of travellers to changes and developments when investigating their effects on network performance. The outputs calculated from these models include estimates of various costs and flows.

On the Right Track

There are one million kilometres of railways in the world servicing over three trillion (1012) passenger-kilometres per year. From urban public transit systems to suburban/ regional railways to high-speed intercity trains, millions of people rely on railways for work, leisure and travel. Therefore, making sure that the trains get the passengers to their destinations safely, affordably, comfortably and on time is not an easy task.

Full contents page of the October 2016 printed issue
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Content from the August issue


Celebrai! Rio de Janeiro is where the parties will be dur­ing August and September, as the Summer Olympic and Paralympic Games converge on this beautiful Brazilian city, which paradoxically will be celebrating winter. The statistics are stunning: more than 10,000 elite athletes from 206 nations will compete for 306 sets of Olympic medals in 39 sports, which will offer an amazing variety of challenges for competitors and much excitement for those of us lounging on sofas. These sports include two new events, rugby sevens and golf. Similarly impressive, more than 4,000 athletes from 56 nations will compete for Paralympic medals in 23 sports. These sports also include two new events, canoeing and triathlon: a thrilling prospect indeed.

Hitchin wins Shaw Prize

This year’s Shaw Prize in Mathematics was awarded to Professor Nigel Hitchin FRS (Savilian Professor of Geometry, University of Oxford) ‘for his far reaching contributions to geometry, representation theory and theoretical physics. The fundamental and elegant concepts and techniques that he has introduced have had wide impact and are of lasting importance.’

Indeed, Hitchin’s profound and imaginative work has had tremendous impact in many different areas of Mathematics, including algebraic geometry, differential geometry, complex analysis, topology, integrable systems, mathematical and theoretical physics.

Mathematics and Financial Markets

The David Crighton Lecture 2016, was delivered by Frank Kelly on Thursday 12 May 2016 at the Royal Society. The following is a brief outline of the lecture.

A substantial proportion of mathematics graduates, at both first degree and doctoral level, enter the financial services sector (Table 1). This is hardly surprising given the apparent importance of the sector to the economy (Figure 1), and the role of mathematical modelling in the valuation of instruments and the assessment of risk. What is striking is that, with some notable exceptions, few mathematicians have been actively engaged in the design of financial markets. This is undoubtedly a serious challenge with parallels from other large-scale complex networks: to design a distributed system, linking self-interested and intelligent agents, so that the outcome is effective and efficient.

Maths in a Twist

It seems such a simple thing. Take a strip of paper, bend it around, and join the ends. You end up with a familiar and not very interesting object – a simple cylinder. It’s no surprise that when we cut the cylinder in half we end up with two, somewhat narrower, cylinders. This is easy to do, and easy to visualise.

But this article is going to be an introduction to topology. In topology we discard the concept of distance and angle. We are allowed to smoosh things about, stretching, squashing, and generally deforming things. No tearing, cutting, and glueing allowed, unless we put things back afterwards.

For these reasons topology has sometimes been called ‘rubber sheet geometry’. Topology does talk about shapes, but it turns out that that description is attractive, catchy, and wrong. We will see why later.

Full contents page of the August 2016 printed issue
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Content from the June issue


The question facing voters on 23 June is ‘Should the [UK] remain a member of the [EU] or leave the [EU]?’ with corresponding answers ‘Remain ...’ and ‘Leave ...’. This format is well designed: clear, concise and impartial. In contrast, regional referendums on Welsh devolution (2011) and Scottish independence (2014) had draft questions ‘Do you agree ...’, which would have led voters towards apparently favoured positions had they not been changed before implementation. However, their simple answers of ‘Yes’ and ‘No’ could still have introduced bias as affirmation is generally deemed more pleasing than negation. These were also the options for nationwide referendums on the European Community (1975) and alternative voting (2011). The current referendum avoids these problems and has fair campaign rules, though extensive feedback arising from media reports, opinion polls and social networks will surely influence the outcome.

The IMA Lighthill Lecture at BAMC

Delivering the 2016 IMA Lighthill Lecture was not only a great honour, but also an opportunity to reflect on the influence of Sir Michael James Lighthill (1924-1998). Of course, he is well known to this audience as the founder of the IMA and Lucasian professor at Cambridge. He is remembered worldwide as a pioneer in theoretical fluid mechanics, especially in aeroacoustics and swimming, which was also his passion in real life.

Personally, I first came across the work of Lighthill, when I was trying to calculate the induced-charge electro-osmotic flow around a polarisable particle and its resulting electrophoretic motion. Half a century later, his seminal paper on the ‘squirming’ motion of micro-organisms provided a useful mathematical framework that could also be applied in this context. Lighthill’s emphasis on broken symmetries in swimming also foreshadowed recent developments in induced-charge electrophoresis.

A Tennis Assignment Algorithm

For some years, I have played social tennis at a local club and have recently organised midweek men’s doubles matches for those who are retired, work part-time or have flexible working arrangements. This used to consist of asking each member of the group about their availability in the coming week, and how much they would like to play, and from their responses, putting together a set of fours using just pen and paper. However as the numbers increased, I started to think about how I could make the process easier and more efficient by writing some code and treating it as an optimisation problem. This article describes how I tackled the problem.

The initial purpose of the algorithm was to automate what I had done manually, by finding a feasible assignment of players to groups across the week and to maximise the number of groups formed. As it is clear that generally there are many possible solutions, the next step was to remove any bias or favouritism in the choice of the groups, by generating all possible feasible and equally-optimal assignments and choosing randomly from them.

Full contents page of the June 2016 printed issue
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Content from the April issue


One of the most enjoyable aspects of being a mathematician in academia is encountering diverse and exciting challenges. One such incident occurred recently when a refrigeration engineer brought two tubes into my office and plonked them on my desk. ‘Can you model intrinsic weight gain and predict time to saturation?’ she asked. As so often happens, my mouth uttered ‘Yes’ with little input from my team and despite my ignorance.

These right annular cylinders were 127 mm long and sealed at both ends, with external diameter 124 mm and bore 70 mm, as illustrated opposite. One of them rattled ominously, though my visitor explained that they were sections of pipe lagging and that the noise was caused by a desiccant. Cold liquid refrigerants pass through the pipes, with the adverse effect of causing costly condensation damage. The engineer had executed the following well designed experiment, which was repeated many times for tubes of various materials and dimensions, in order to address this problem.

Raising the Profile of Black Mathematicians

According to Professor Rosina Mamokgethi Setati-Phakeng, the first Black South African female to get a PhD in Mathematical Education:

Being the first is not something to be proud about, it is a calling to ensure you are not the last.

My name is Dr Nira Chamberlain and I became the first Black Mathematician to be referenced in Who’s Who, which is a lead­ing source of biographical data on more than 33,000 influential and distinguished people from around the world. Published an­nually since 1849, I am only one of approximately 30 mathema­ticians referenced in the 2015 edition of this book. Inclusion has therefore come to carry a considerable level of prestige.

Small Worlds by Design

Connecting all the members of a large group involves many exchanges; think of it as clinking glasses at a social gathering. Allowing second-hand exchanges/clinks where the group splits into subgroups who clink glasses, and then representatives from each subgroup clink with each other – second-hand clinking – can decrease the number of clinks. This paper finds the optimal number of subgroups necessary to minimise the total number of clinks. It also looks at what is optimal if there is third and higher level clinking. Examples are given of such problems arising in parallel computer systems, social media, communication systems, and sports tournaments.

Historical Notes: A Mathematical Inscription from Ancient Pergamon

During the latter part of the 18th century, the wonderfully named polymath Comte Marie-Gabriel-Florent-Auguste de Choiseul-Gouffier, French ambassador to the Ottoman Empire based in Constantinople, embarked on an extensive series of travels around the Aegean and Asia Minor. Writing up his journeys and observations in the similarly wonderfully named Voyage Pittoresque de la Grèce, he copied the details of a particular inscription, written in Greek, found at Pergamon and dating from the 2nd century CE during the Roman Antonine period. This inscription is the only ancient one, known to me, that contains any mathematics. Sadly the stone itself would now appear to be lost.

Full contents page of the April 2016 printed issue
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Content from the February issue


Let me start by congratulating Professor Chris Linton on his recent appointment as IMA President and wish­ing him well during his term in office: the IMA will undoubtedly prosper under his leadership. His predecessor, Professor Dame Celia Hoyles, was an excellent president who achieved much over the past two years.

Among her many contributions, Celia was guest editor with Professor Richard Noss for December’s special issue of Mathematics Today on the theme of education, surely one of our most important professions. I hope that you en­joyed reading the wide selection of entertaining articles by international authors, which clearly demonstrate the fluidity of mathematics and the adaptability of teaching to incor­porate new technology as a means of enthusing and inspir­ing our students. For those interested, IMA Councillor and former Honorary Secretary, Professor Nigel Steele, and his colleague Matthew Bulmer (NCTL) will present a talk on Training Mathematics Teachers at the Mathematics 2016 conference in London on 17 March.

The 2015 ECM Autumn Conference at Bath

On Saturday 14 November, more than 80 mathematicians in the early stages of their career gathered together at the University of Bath to hear and talk about promising areas of mathematics, and the challenges that young mathematicians face in those fields. The day was a big success despite the unfortunate weather, and it gave ample opportunities to everyone who attended to network with their peers from different universities and organisations.

How to Make the Perfect Pancake

We explore the cooking of pancakes using a combination of kitchen experiments and mathematical theory. The properties of a pancake are characterised in terms of a dimensionless geometrical measure of aspect ratio (I1) and the baker's ratio that describes the mass ratio of water or milk to flour (I2). The patterns on the top and bottom of pancakes are analysed in a kitchen study and explained in terms of how the vapourised liquid in the batter escapes. We determine the properties of a perfect pancake.

Pancakes are a starch-based comestible created by pouring batter onto a hot solid surface and cooking until solid [1]. They come in a lot of different shapes from the large, thin, circular French crêpes to small, thick, circular drop scones from Scotland or ball-shaped æbleskiver found across Scandinavia.

The Participation of Girls in Further Mathematics
Further Mathematics (FM) is a qualification designed to broaden and deepen a student’s mathematical knowledge, and can be taken to either AS level or A-level alongside AS or A-level Mathematics courses. It is valuable for a number of reasons, including:
  • the increased time spent engaging with mathematics and developing greater fluency;
  • the study of important topics in pure mathematics not covered at A-level, such as complex numbers and matrices, that are essential for anyone going on to study maths, physics or engineering;
  • the opportunity to study a broader range of applications of mathematics;
  • the development of increased confidence and resilience in tackling demanding mathematical problems.

Full contents page of the February 2016 printed issue
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