Brian Cox

The physicist Brian Cox, presenter of the BBC2 series Wonders of The Solar System has teamed up with the Big Bang science fair to launch the Big Bang Lesson. Brian will visit one school somewhere in the UK and deliver a lesson based around the solar system. The class is designed to inspire pupils and generate interest in science, technology, engineering and maths. Students aged 11-18 can enter their school into the draw through Facebook or the Big Bang website. One UK school will be picked at random to receive the lesson in November.

Bad statistics can mislead, and who'd know this better than mathematicians? It's ironic, then, that mathematics itself has fallen victim to the seductive lure of crude numbers. Mathematicians' work is being measured, ranked and judged on the basis of a single statistic: the number of times research papers are being cited by others. And mathematicians are not happy about it.

Is this good maths?

Like any other area in receipt of public money, mathematical research needs to be accountable. A reasonable way to judge the quality of research is the impact it has on future research: ground-breaking work will be heavily discussed and built upon, and mediocre work largely ignored. Traditionally, the reputations of individual researchers, institutions, or research journals have hinged on the opinions of experts in the field. The rationale behind using citation statistics is that bare numbers can overcome the inherent subjectivity of these judgments. In a competitive world it's the bottom line that should do the talking.

Bottom lines are crude, however, and summary statistics open to misuse. A whiff of scandal floated through this year's International Congress of Mathematicians, when the mathematician Douglas N. Arnold (president of the Society for Industrial and Applied Mathematics) exposed what appears like a blatant example of citation fraud. It involves the International Journal of Nonlinear Science and Numerical Simulations (IJNSNS) and a summary statistic called the impact factor.

The impact factor of a journal measures the average number of citations per article in the journal, but only taking into account citations from the current year to articles that have appeared in the previous two years. So old citations don't count and neither do citations to articles that are older than three years.

IJNSNS has topped the impact factor chart for applied maths journals for the last four years by a massive margin. In 2009 its impact factor was more than double that of the second in line, the esteemed Communications on Pure and Applied Mathematics (CPAM). A panel of experts, however, had rated IJNSNS in its second-to-last category: as having a "solid, though not outstanding reputation". In the experts' opinion IJNSNS comes at best 75th in the applied maths journal rankings, nowhere near the top.

There are some easy explanations for this mis-match between the impact factor chart and expert opinion. A closer look at citation statistics shows that 29% of the citations to IJNSNS (in 2008) came from the editor-in-chief of the journal and two colleagues who sit on its editorial board. A massive 70% of citations to IJNSNS that contributed to its impact factor came from other publications over which editors of IJNSNS had editorial influence. An example is the proceedings of a conference that had been organised by IJNSNS's editor-in-chief Ji-Huan He. He controlled the peer review process that srcutinised papers submitted to the proceedings.

Another striking statistic is that 71.5% of citations to IJNSNS just happened to cite articles that appeared in the two-year window which counts towards the impact factor (the 71.5% is out of citations from 2008 to articles that have appeared since 2000). That's compared to 16% for CPAM. If you use a five-year citation window (from 2000 to 2005) to calculate the impact factor, IJNSNS's factor drops dramatically, from 8.91 to 1.27.

The conclusions from this are obvious: cite your own journal as often as possible (with citations falling in the two-year window) and make sure that authors who fall under your editorial influence do the same, and you can propel your journal to the top of the rankings.

Libraries use impact factors to make purchasing decisions,
but mathematicians are judged by them too.

What's worrying is that impact factors are not just being used to rank journals, but also to assess the calibre of the researchers who publish in them and the institutions that employ these researchers. "I've received letters from [mathematicians] saying that their monthly salaries will depend on the impact factors of the journals they publish in. Departments and universities are being judged by impact factors," says Martin Grötschel, Secretary of the International Mathematical Union, which published a highly critical report on citation statistics in 2008.

Grötschel dismisses the blind use of impact factors as "nonsense" and not just because they are open to manipulation. For mathematics in particular, the two-year window that counts towards the impact factor is simply too short. There are examples of seminal maths papers that didn't get cited for decades. In fact, scouring 3 million recent citations in maths journals, the IMU found that roughly 90% of all citations fall outside the two-year window and therefore don't count towards the impact factor. This is in stark contrast to faster moving sciences, for example biomedicine, so using impact factors to compare disciplines presents mathematics in a truly terrible light.

Another confounding factor is that papers may get cited for all the wrong reasons. As Malcolm MacCallum, director of the Heilbronn Institute for Mathematical Research, pointed out at a round table discussion at the ICM, one way of bumping up your citation rates is to publish a result that's subtly wrong, so others expose the holes in your proof, citing your paper every time. Malicious intent aside, someone might cite a paper not because it contains a ground-breaking result, but because it gives a nice survey of existing results. If on the other hand your result is so amazing that it becomes universally known, you might lose out on citations altogether — few people bother to cite Einstein's original paper containing the equation E=mc² as the result and its originator are now part of common knowledge.

The list of impact factor misgivings goes on (you can read more in the IMU report on citation statistics). The fact is that a single number cannot reflect a complex picture. With respect to manipulation, Arnold points to Goodhart's law: "when a measure becomes a target, it ceases to be a good measure". What's more, no one knows exactly what the impact factor is supposed to measure — what exactly does a citation mean? As a statistical quantity the impact factor is not sufficiently robust to chance variation. As the IMU report points out, there's no sound statistical model linking citation statistics to quality.

How, then, should mathematical quality be measured? Mathematicians themselves aren't entirely in agreement on how big a role, if any, citation statistics should play, or even whether things should be ranked at all. Everyone agrees, however, that human judgment is essential. "Impact factors — we cannot ignore them, but we have to interpret them with great care," says Grötschel, the IMU Secretary. The IMU, together with the International Council of Industrial and Applied Mathematics, has set up a joint committee to come up with a way of ranking journals that might involve human judgment and statistics.

With their fight against the mindless use of statistics mathematicians will do a service not just to themselves. "Some of [our work on this] has very broad applications in other sciences," says Grötschel. "It's very important that mathematicians are at the forefront of this issue."

Further reading

• Read Douglas N. Arnold and Kristine K. Fowler's article Nefarious Numbers.
• The use of citation statistics is explored in this IMU report.

For the latest episode of the Math/Maths podcast Peter Rowlett interviewed a very tired and exhausted Plus, reporting from the ICM in Hyderabad, India. It's nice to be on the other side of the microphone for a change! The podcast also explores Roberto Carlos' free kick; A New Kind of Baseball Math; More on P !=NP; The #mathgeek experiment; Clustered Networks; measuring physical constants; testing string theory; Twitter Venn; Mangahigh; and more.

The Math/Maths Podcast: 5136 miles of mathematics is a weekly conversation about mathematics between the UK and USA from Pulse-Project.org. Peter Rowlett in Nottingham calls Samuel Hansen in Las Vegas and the pair chat about math and maths that has been in the news, that they've noticed and that has happened to them.

Many things in life are more than the sum of their parts. Whether its the behaviour of crowds of people, flocking birds or shoaling fish, the unpredictable patterns of the weather or the complex structure of the Internet, it's often the interaction between things, rather than the things themselves, that generates complexity.

It's a challenge to science, whose traditional approach of taking things apart and looking at the individual bits doesn't work when faced with emergent complexity. But there are mathematical techniques to understand this phenomenon. The Living in a Complex World website, originally launched to accompany an exhibit at the Royal Society Summer Science exhibition, explores complexity in the real world and has some great factsheets looking at the maths used to understand it. It's well worth a look!

Here are some pictures from the ICM 2010:

Plus headed for world domination.

Well, maybe not quite ... it's a panel discussion on popularisation of maths. (Thanks to Jaime Carvalho e Silva for both of these photos.)

Plus with Cédric Villani.

3000 mathematicians trying to have dinner.

3000 mathematicians trying to have lunch.

3000 mathematicians trying to catch a hotel bus.

Plus with Christian Schlaga, Germany's acting ambassador to India. It's a long story, but basically Plus ended up with a sculpture of the Berlin bear (with a maths design) that had been presented to Schlaga at the German embassy's reception at the ICM.

The old town of Hyderabad

What would you think if the nice café latte in your cup suddenly separated itself out into one half containing just milk and the other containing just coffee? Probably that you, or the world, have just gone crazy. There is, perhaps, a theoretical chance that after stirring the coffee all the swirling atoms in your cup just happen to find themselves in the right place for this to occur, but this chance is astronomically small.

Cédric Villani, Institut Henri Poincaré
Fields medallist 2010.

The fact that such spontaneous separation never occurs in practice is an illustration of a deep physical law: it says that the entropy of a system, a measure of its complexity, almost always increases as time passes. When you first pour the milk into your coffee, for a split second milk and coffee will be neatly separated, but soon the milk disperses. The mixture of milk and coffee becomes more and more complex, until it reaches an equilibrium when both are completely mixed up and complexity is at its peak.

In the late 19th century the physicist Ludwig Boltzmann studied this phenomenon, looking at what happens when a gas is released into a room from a bottle. He came up with an equation describing the evolution of this process — the change over time — in terms of how individual atoms collide and other forces acting on the gas. His calculations showed that indeed entropy doesn't decrease. The atoms of gas start out in an ordered state — all sitting in the bottle — and end up in a state of maximum complexity, dispersed throughout the room.

Intriguingly, this result meant that there is what physicists call an arrow of time, something that isn't inherent in classical Newtonian physics. If someone showed you a movie of a billiard ball rolling across the table, then you wouldn't be able to tell if the movie was being played forwards or backwards: it's just as likely that the ball rolls one way as it is to roll in the opposite direction. If, however, someone showed you a movie of a coloured gas dispersed in a room suddenly entering a bottle, you'd know that something's wrong. The movie is being played backwards. Since the interaction of individual atoms was described in terms of Newtonian laws (which don't have a preferred direction of time), this emergence of an arrow of time created some headache for physicists. Debate on the arrow of time issue continues to this day. As the mathematician John von Neumann once put it, "nobody knows what entropy really is, so in a debate you will always have the advantage".

Villani, however, does understand was entropy is. One question that has until recently remained open was how fast the entropy of a system increases. Cédric Villani received his Fields Medal for developing rigorous mathematical techniques that provide an answer. They show that while entropy never decreases, it sometimes increases faster and sometimes slower. Based on this work, Villani developed a general theory, hypercoercivity, which applies to a broad set of situations.

Villani also used his understanding of entropy to explain a phenomenon that had puzzled physicists for 60 years. Back in the 1940s, the Soviet physicist Lev Davidovich Landau claimed that plasma, a form of matter similar to gas, spreads and reaches its equilibrium state without increasing its entropy. Landau argued that unlike gas, whose approach to equilibrium is driven by particle collisions (which also lead to a loss of order), plasma reaches equilibrium through a decay in its electric field. Landau made some progress on proving his claim, but despite hard work by many physicists over 60 years, it wasn't until Villani's arrival on the scene that Landau was proved right.

All this might sound somewhat esoteric, but Villani's deep understanding of entropy has direct consequences for real-life problems. In the 18th century the French mathematician Gaspard Monge started to think about how to transport goods to various places in an efficient way. For example, you might want to distribute a bunch of letters sitting in a post office to the various addresses in a way that minimises transport cost. Villani and his colleague Felix Otto made a crucial connection between this problem and gas diffusion, by drawing the following analogy: loosely speaking, the initial state (all letters in the post office) corresponds to the ordered state of gas sitting in a bottle, while the end state (letters delivered) corresponds to a state where the gas has dispersed. To any configuration of dispersed gas particles you can assign a cost by seeing how far the particles had to travel from the original ordered state. Using this analogy and their understanding of entropy, Villani and Otto made important contributions to optimal transport theory.

So next time you have a cup of milky coffee or receive a letter, stop to think that a deep understanding of either, or even better a combination of both, could have earned you a Fields medal.

You can find out more about Villani's work in this excellent description on the ICM website, on which this blog post was based. To find out more about the Boltzmann equation and optimal transport problems, read the Plus article Universal Pictures.