More than 120 years ago at the International Congress of Mathematicians, German mathematician David Hilbert presented a list of 23 then-unsolved problems. Since then, mathematicians the world over have worked to solve them. Today, Lei Wu is tackling the sixth on the list.
Hilbert’s sixth problem seeks to axiomatize the branches of physics in which mathematics are prevalent, says Wu, an assistant professor of mathematics. In other words, he is attempting to reveal their inherent truth.
“It tries to put physics into a solid mathematical foundation,” he explains. “If you just use a few basic laws like Newton's laws or some basic quantum mechanics laws to describe all the other phenomena … you need to justify whether this is really possible or just somehow an illusion. So, from the mathematical side, we want to rigorously justify: Yeah, this is doable.”
Take as an example kinetic theory, which attempts to describe the dynamics of a large number of particles in a space. A researcher can utilize several mathematical approaches to understand the behavior of these particles. Depending on the particular scenario, some methods are better than others, says Wu. He wants to help researchers choose.
One approach, he says, is Newton’s second law, through which a scientist might track the position and the velocity of each particle of air in a room. However, a micro-scale model with such large numbers would require a supercomputer, and the size of the room could determine that even a supercomputer might not be the answer: “It looks good, but it's not practical,” says Wu.
Another possible approach is the use of thermodynamics or fluid mechanics, which focus on factors such as temperature, pressure and velocity to find a statistical average of the properties, rather than focusing on the behavior of a single particle. This type of macro-scale modeling, however, might not be precise enough.
Kinetic theory invites the question of how to connect the two approaches: Do they describe the same thing? Is it possible that just one is correct? Wu seeks to identify the criterion a researcher can follow to determine under which regime, or circumstance, to use which approach. He focuses on how to study this problem in a bounded domain—that is, when the particles in question come into contact with a natural barrier, such as water filling a drinking glass, flows of air particles passing an airplane wing or neutrons colliding in a nuclear reactor.
“We want to study this kind of phenomenon and give a quantitative description of what happened exactly,” he explains. This helps him justify which approach is better for a particular situation.
The right decision matters. Levels of difficulty vary. Computers used for numerical simulations come with time and memory costs that depend on the chosen model. A researcher ultimately should choose the most proper and precise approach possible, says Wu.
“If we use fluid dynamics … and then we know that it is already precise enough in certain regimes, then just use it. We don't need other more precise models. But when in a different regime, we say, well, the kinetic equation is more proper … then you have to choose that. Although it is very difficult, it is still the more precise one. We want to tell you which regime you should use to be more economic and to be more efficient,” he explains.
This work is funded by the National Science Foundation.