Discovery Files

Mathematician tries to solve wave equations

Breakthrough Prize winner seeks further insights into fluid dynamics

Wave equations help describe waves of light, sound and water as they occur in physics. Also known as partial differential equations, or PDEs, they have valuable potential for predicting weather or earthquakes, or certain types of natural disasters. For example, during the late stages of a tsunami, they could help forecasters calculate when it will hit land. "PDEs are a big reason why math is useful," says Terence Tao.

Tao, a professor of mathematics at UCLA, is interested in the theoretical side of these equations, seeking to discover with computer algorithms whether they can behave in a way that typically is the opposite of what occurs in the real world. He wants to see whether they can "exhibit blowup," or, essentially, explode.

To explain what he means by this, the National Science Foundation (NSF)-supported scientist suggests picturing what happens to ripples on the surface of a lake or pond. Usually, they gradually disperse and disappear. He, on the other hand, is trying to create the opposite effect, starting with still water that gathers force and ends with a blast.

"Imagine a whole bunch of ripples in concentric circles converging to a single point and exploding," he says "The initial conditions are smooth and placid, but as you run wave equations, they could spontaneously create oscillations, or rogue waves."

This could mean unexpected waves that rise from a calm surface, an occurrence that occasionally appears in nature, although "we don't know if they are spontaneously formed or whether there is an external force creating them, like a tsunami or an unusual weather pattern," Tao says.

His experiments involve trying to solve a series of wave equations, testing whether they are actually possible. He and other mathematicians work with dozens of equations that cover a wide range of possible scenarios from the basic laws of physics--one for every type of fluid and every situation, for example, deep water, shallow water, etc.

"You take an equation and either prove its regularity," meaning they start smooth and end up that way again, "or show that they can do the opposite, start smooth and become faster and faster in amplitude," he says.

"Sometimes when you run these equations they will predict your fluid will reach infinite velocity, but this is impossible, meaning at some point your math equations will break down in the real world," he adds. "Sometimes they give results that are not physical, that is, they don't make any physical sense, meaning they aren't trustworthy."

Tao is an Australian-American mathematician who began learning calculus at age 7, at which age he began high school, earned his doctorate from Princeton University at age 20, when he joined the UCLA faculty, and became a full professor at 24.

He has been the recipient of several NSF grants since 1997 totaling more than $1.3 million, the most recent awarded in 2013. He also was awarded NSF's prestigious $500,000 Alan T. Waterman Award in 2008, which recognizes an outstanding young researcher in any field of science or engineering supported by NSF.

Most recently he won the $3 million Breakthrough Prize in Mathematics, launched by Facebook founder Mark Zuckerberg and Russian entrepreneur Yuri Milner, which recognized him for major advances in the field, and for contributing to communicating the importance and excitement of mathematics to the general public.

Tao has developed insights into a range of different mathematical areas, including harmonic analysis, combinatorics (the branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those of graph theory), and number theory.

He has made significant advances in problems such as Horn's Conjecture, which he and Allen Knutson, professor of mathematics at Cornell University, showed can be reduced to a geometric combinatorial configuration known as a "honeycomb;" honeycombs are connected to several other areas of mathematics, including representation theory, algebraic geometry and abstract algebra.

Also, his analysis of the Schroedinger equation, a PDE that describes how the wave function of a physical system evolves over time, a central element of quantum mechanics, produced new techniques for solving nonlinear partial differential equations.

He and others are very interested in solving the Navier-Stokes equations, which are among the most difficult tackled by mathematicians, and deal with the motion of fluid substances. Understanding them could help with modeling weather, ocean currents, the flow of water through a pipe, air around an airplane wing, and even blood through veins and arteries.

"This is one of the basic equations we use," he says. "We ask, if you start with an initial condition of fluid that is nice and smooth, and let time evolve, can the solution to these equations ever blow up? It shouldn't happen, but we've not been able to settle the question one way or the other. We don't know."

One of the things he is trying to prove is whether, "if you design a special set of initial conditions, you can create a solution to Navier-Stokes where it does become infinite over time," he says. "It would be the reverse of a stone being thrown into a pond, and ripples. I want to do it in a way where it starts smooth, you get ripples and you end with an infinitely fast splash. We don't see that happen in the real world. It is a rare occurrence in the math world, but I think they do exist, theoretically, and that is what I am trying to find."

If mathematicians could establish a blowup for Navier-Stokes, the result would have important implications for the foundations of fluid mechanics, "in that one occasionally needs to replace the Navier-Stokes equations by some more sophisticated refinement when the former equations are predicting a physically impossible blowup," he says.

"But in pure mathematics, we never really know where the applications are going to show up when working on foundational issues," he adds. "For instance, [Bernhard] Riemann worked on an abstract theory of curved space without any notion that Einstein would one day need them for his theory of general relativity; it's what Eugene Wigner famously called 'the unreasonable effectiveness of mathematics in the physical sciences.'"