MathWorld Headline News

Smale's 14th Problem Solved

By Eric W. Weisstein

February 13, 2002--In a paper appearing in this month's Foundations of Computational Mathematics, W. Tucker of Cornell University has become the first person to lay to rest one of Steven Smale's challenging math problems for the twenty-first century.

In 1998, mathematician and Fields medalist Steven Smale published a list of problems that would prove challenging to mathematicians in the upcoming century (Smale 1998, 2000). This list was proposed in the spirit of Hilbert's problems, a list of problems put forward by mathematician David Hilbert in 1900, whose solutions Hilbert envisioned would lead to significant furthering of various disciplines of mathematics.

With the publication of Tucker's paper, the 14th problem on Smale's list has become the first to be cracked. Smale's 14th problem asks if the structure of the solution to the so-called Lorenz equations is that of a strange attractor. The Lorenz attractor is the solution space that arises in a simplified system of equations describing the 2-dimensional flow of fluid of uniform depth in the presence of an imposed temperature difference and with gravity, buoyancy, thermal diffusivity, and kinematic viscosity (friction) taken into account. In the early 1960s, Lorenz accidentally discovered that this system exhibits chaotic behavior when he found that, for a simplified system, periodic solutions became larger above a certain critical cutoff value. He also found that vastly different results were obtained for very small changes in the initial values. This represented one of the earliest discoveries of the so-called butterfly effect.

While the Lorenz attractor has been intensely studied for many years, it has previously proved impossible to show that the properties of this solution are exactly those of a mathematical structure known as a strange attractor. Tucker has now laid this nearly 40-year-old problem to rest with a technical proof that makes use of a combination of normal form theory and validated interval arithmetic.

It remains to be seen if any other of Smale's problems can be dispatched as quickly or, like a number of Hilbert's problems, will remain open long into the twenty-second century. References