The sea was filled with yellow ducklings, blue turtles, green frogs and red beavers on January 10, 1992. A cargo ship fell overboard a container with 29,000 plastic toys for the bathtub during a violent storm in the North Pacific, just halfway between Asia and America. Seven months after the accident, they began to meet hundreds of rubber dolls off the coast of Sitka, Alaska, not far from another 61,000 Nike shoe spill that occurred two years earlier. To the American oceanographer Curtis Ebbesmeyer It occurred to him then to be attentive for years to the sightings of those drifting objects, with the aim of learning to predict marine currents. Four Spanish mathematicians, faced with another monumental problem, have rebound solved the enigma of rubber ducks floating in the Pacific: it was impossible to predict on which beach they would appear. It seems like fun, but the research is published in one of the best scientific journals in the world, *PNAS*, for its potential implications for humanity.

An American foundation, the Clay Institute of Mathematics, announced in 2000 that it would give a million dollars to whoever solved any of the so-called “seven millennium problems“Devilishly complex mathematical riddles. One of them has to do with the Navier-Stokes equations, which describe the motion of liquids and gases. They were formulated between 1821 and 1845 by the French mathematician Claude-Louis Navier and by the irish physicist George Stokes. Taking into account factors such as temperature, viscosity, and the initial velocity of a fluid, the equations calculate its velocity at a later time. Almost 200 years after its formulation, it is unknown if the solutions to these formulas always have a certain regularity or can have an explosion, a sudden change in behavior. Finding out is worth a million dollars.

The math Eva miranda She tells that she was bored on the Madrid-Barcelona train one day in February 2019 when, browsing the social network Twitter, she found out that the Australian researcher Terence Tao | had published on his blog the latest details of your battle to solve that millennium problem. Tao was a child prodigy of mathematics. At the age of 14 he began his university degree in Science. At the age of 20, he had already obtained a doctorate in mathematics at Princeton University (USA). At the age of 31, he received the Fields medal, one of the most prestigious awards in the discipline. And now, at 45, they still call him “the Mozart of mathematics”.

Tao announced in 2014 that he intends to abstractly simulate a kind of computer made with water, to force the liquid to accumulate energy until a sudden change is detected with the Navier-Stokes equations. Miranda, a professor at the Polytechnic University of Catalonia born in Reus (Tarragona) 47 years ago, the light bulb went on on the train and she called three colleagues to propose an idea.

The four mathematicians —Robert Cardona, Eva Miranda, Daniel Peralta and Francisco Presas— announced this wednesday who have managed to design an abstract water machine for the first time. The researchers have been based on a Turing machine, a device that receives a sequence of numbers with a binary code, ones and zeros, and generates a result, also expressed in ones and zeros, after applying certain rules. The water machine of the four mathematicians takes as input a point in space and offers as a result the point to which the fluid has moved.

Miranda explains that his machine shows that the turbulent behavior of fluids is an “undecidable” problem: mathematics falls short of solving it. It is not that mathematicians are clumsy or it has nothing to do with the famous unpredictability of chaos theory (the flapping of a butterfly generates a tornado), it is that no algorithm can say that a fluid will pass through a point in a certain time. “We are the first to show that you can’t find rubber ducks, assuming they move in three dimensions,” Miranda emphasizes. “It is as if I throw a message of love in a bottle into the sea. It will follow a path and after a while it will be elsewhere. What we have shown is that we cannot predict where it will be, so it is better to send a wasap ”, jokes the mathematics.

Some rubber toys from the 1992 accident appeared years or even decades later on the shores of half the world, according to the BBC. in his documentary series *Blue Planet 2**.* Some made it to Alaska, some to Australia, some to Japan, and some even crossed the Arctic and went from the Pacific Ocean to the Atlantic. Eva Miranda recalls that, shortly before the duckling dumping, the American physicist Cris Moore you already wondered if the fluids are complex enough as to do all the possible operations of a computer. “We show that it is,” Miranda sentenced.

Terence Tao applauds the work of his four Spanish colleagues. “More than the solution, it is an evidence of the difficulty of the problem [del millón de dólares]”, He explains to EL PAÍS. The new water machine does not apply to the flat three-dimensional space in which we live, but to a simplified curved version, Tao clarifies. “But it does show that fluids can become so complex in those curved spaces that, in a way, they can behave like a computer, so you can program a fluid to do anything that can be programmed in a computer,” explains the researcher. Australian, from the University of California, Los Angeles. “I believe that the same type of phenomenon also occurs in our flat three-dimensional world, which would rule out many types of approaches to solving the problem of the regularity of the Navier-Stokes equations, which are based on showing that fluids obey in certain ways simple ”, he adds.

Tao’s hypothesis is that the Navier-Stokes equations will not present a global regularity, but rather “will explode”. This does not mean that a tsunami appears suddenly in the ocean of the real world, but that under certain conditions these equations do not serve to describe the complexity of fluids. Miranda compares the path of a fluid particle to the stroke of a pencil on paper. “To explode would be for someone to nudge you and move the pencil. In the drawing you would see a discontinuity, a very strange behavior, like going to infinity ”, the researcher illustrates.

Miranda believes that if the Navier-Stokes equations explode it would be “a real revolution”, because it would suggest that mathematical models are imprecise to predict weather, sea level rise or the behavior of other essential viscous fluids, such as blood. human and oil.

The abstract water machine of Miranda and his colleagues does not use the Navier-Stokes equations, but an earlier version, formulated in 1755 by the Swiss mathematician Leonhard Euler to describe the movement of ideal fluids, without viscosity. The solutions offered by the machine do not show sudden jumps. “Our research does not serve to demonstrate the explosion of the Navier-Stokes equations,” says the professor.

Eva Miranda, Daniel Peralta and Francisco Presas are members of the Institute of Mathematical Sciences (ICMAT), a research center of excellence in Madrid. Presas, 46, was interviewed by this newspaper when he was 21, after winning a Seat Ibiza in the contest entitled *College of the year*, sponsored by the automobile company. “I am good at mathematics,” he declared then. Today is an international figure in geometry, like Miranda and Peralta. Robert Cardona does his doctoral thesis at the Polytechnic University of Catalonia.

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Eddie is an Australian news reporter with over 9 years in the industry and has published on Forbes and tech crunch.