19/12/2018

Unveiling the mysteries of nature and computing.

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

As I've mentioned before, in 2000 the Clay Mathematics Institute offered a prize for solving seven mathematical problems, the so-called Millennium Problems: one million dollars for whoever solved each problem. I've already written about five of these problems here. I've left the two most directly related to the applications of mathematics for last.

The Navier-Stokes equation describes the behavior of viscous fluids, constituting a mathematical model for many real-life phenomena: blood flow, ocean currents, airflow around an aircraft, weather, and many others. If we knew how to solve the equation mathematically, it would have important (and very lucrative!) applications in countless fields. The problem is that we don't even know if the equation has solutions.

Claude-Louis Navier (1785 – 1836) was a French engineer and mathematician, renowned for his bridge construction until the collapse of one of his projects in 1824 tarnished his reputation. This did not prevent him from receiving numerous honors: he is one of 72 leading figures (all men) in French science and technology whose name is engraved on the Eiffel Tower in Paris.

Stokes also received important distinctions and presided over the Royal Society of London. He is well known to mathematics and engineering students worldwide for the so-called Stokes' Theorem of vector calculus. This is quite ironic, because this theorem is attributed to the Irish physicist Lord Kelvin (1824–1907): Kelvin explained the result to Stokes in a letter, but Stokes contributed so much to popularizing it that it ended up bearing his name.

The seventh millennium problem, which I consider the most fascinating, comes from computing. With the advent of computers, it has become clear that some problems are more difficult (that is, more time-consuming) to solve through computing than others.

Determining whether a given number is prime is computationally easy: in 2002, Indians Manindra Agrawal, Neerak Kayal, and Nitin Saxena discovered a quick method to solve it. We say that this is a P-type problem.

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