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6 February 2014updated 28 Jun 2021 4:46am

Maths is all Greek to me: how language barriers influence mathematics

The Navier-Stokes equations, which describe how fluids such as air and water flow, may finally have been proved to work in every situation.

By Michael Brooks

Here’s a little-known fact: to get a PhD in maths from Harvard, you need to be a language buff. The university points out that “almost all important work” is published in French, German, English or Russian, and so “every student is advised to acquire an ability to read mathematics in French, German and Russian”. This makes it sound optional, but if you want the PhD you have to pass a two-hour written exam in two of these three languages.

Even so, we now have a mathematical pile-up at the language barrier. The Kazakh mathematician Mukhtarbay Otelbayev says he has solved a problem that has stumped mathematicians for almost 200 years. If he is right, he can claim a million-dollar prize from the Clay Mathematics Institute in Providence, Rhode Island, which has identified his topic as one of the seven most important open problems in the subject. Unfortunately, Otelbayev published his paper in Russian, and those looking to verify his claim can’t find mathematicians linguistically skilled enough to make sense of Otelbayev’s argument.

It concerns a piece of work known as the Navier-Stokes equations. First written down in 1822, the equations have solutions that describe how fluids such as air and water flow in any given situation. Engineers, meteorologists and oceanographers use these equations – and their solutions – every day. However, we have never had proof that the equations can be solved reliably in every situation. This is what Otelbayev claims to have found.

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The discovery isn’t just of academic interest. We want to know that we can trust the Navier-Stokes equations in all kinds of circumstances. These equations allow us to understand phenomena such as the polar vortex – the mass of swirling air currents that brought extreme freezing temperatures to North America early in January. They also describe what happens to the oceans when carbon-dioxide concentrations rise; so far, the calculated effects include a weakening of the Atlantic Meridional Overturning Circulation. This is the network of currents that, among other things, creates the Gulf Stream, which keeps the UK’s weather reasonably mild.

Flowing fluids are extremely complicated systems and are often hypersensitive to small changes in composition. This is why it matters so much that the EU reached a legally binding agreement to reduce member-nations’ CO2 emissions.

Consider the effect of carbon-dioxide emissions on “clear-air turbulence”. This is caused by interactions between layers of air travelling at different speeds. Raise the concentration of carbon dioxide, and the solutions to the Navier-Stokes equations tell us that the turbulence moves to more northerly latitudes and higher altitudes. This will put it directly in the path of transatlantic flights. Passengers aren’t the only ones who’ll be made uncomfortable by the change: engineers are concerned that if turbulence becomes more intense, it could shorten the life of aircraft.

We could alter the transatlantic routes but that would use extra fuel and increase carbon emissions. And these are going up anyway, which, the Navier-Stokes equations could tell us, is likely to create still more problems. At least we think it could – but to make sure, we need to find a competent Russian-speaking mathematician. Perhaps Otelbayev should offer to split the Clay Institute cash: after all, those PhD students have to earn back their Harvard tuition fees somehow.