Matthew Blair, professor in the Department of Mathematics and Statistics at The University of New Mexico recently received a Frontiers of Science Award at the inaugural International Congress on Basic Science. This congress took place in Beijing during this summer.

The award was given to Blair and his coauthor, Christopher Sogge, a professor at Johns Hopkins University, to recognize the contribution of their research paper titled, “Logarithmic improvements in Lp bounds for eigenfunctions of the critical exponent in the presence of nonpositive curvature.” The paper was published in Inventiones Mathematicae, a top-ranked mathematics journal. Sogge traveled to Beijing for the congress and formally accepted the award on behalf of both of them.

“I fondly remember one of the eureka moments we had in the process, and from there all the other elements of the solution fell into place quite quickly. Given the struggle we went through, it meant a lot to be recognized with the award. At the time of writing, I felt it had been the best paper I have been involved in and I still believe that today,” expressed Blair. 

The paper concerns so-called “standing waves,” waves that oscillate periodically in time about a fixed profile. They are analogous to solutions of the stationary Schrödinger equation appearing in quantum mechanics. It turns out that they provide a lens into much richer wave phenomena, i.e. the more researchers understand about standing waves, the more they can understand about wave propagation altogether. This is because in many cases, there are mathematical models that allow for a wave to be viewed as a sum of these standing waves. In order to take advantage of this, it is significant to study the behavior of these standing waves as they resonate at increasingly large frequencies.

A familiar example of these standing waves comes from vibrating strings, where they are often called “harmonics,” something familiar to those who play stringed instruments. Blair continues to explain, “As a teenager, my guitar teacher taught me how to tune my guitar by touching two different vibrating strings in just the right spots so that they should (roughly) resonate at the same frequency. If my ear detected a difference in the tone, I would know to adjust the tuning pegs to get the strings back in tune. It was not a perfect method of tuning but turned out to be convenient when I needed to make a quick adjustment.

"Later when I was working towards my BS, I took a class in partial differential equations that went through the mathematical theory of these harmonics. It was an eye-opening moment that showed me how mathematical theories could explain a practice that seemed very separated from math when I first learned it.”

The paper concerns standing wave phenomena that are much richer than vibrating strings. While vibrating strings are essentially a 1-dimensional model, there are higher dimensional models as well, such as vibrating drumheads and plates, and their work considers cases that are closely related to these. In higher dimensions, the amplitudes of these standing waves may get increasingly large as the frequency of vibration increases. Moreover, the underlying geometry of the system affects how large these amplitudes can be, for example, the standing waves generated by a circular drum may behave much differently than those on a square drum.

“Our work considered a means of quantifying the growth of such amplitudes as the frequency increases, known mathematically as an Lp norm. This entails averaging the amplitude in a certain way to get a sense of how large the amplitude may be. We considered particular circumstances where the underlying geometry of the system yields paths of least action which behave chaotically. We showed that in such cases, the Lp norms of the amplitudes of standing waves cannot grow as fast when compared to certain geometries that allow for paths of least action which are very stable and predictable,” said Blair.

The research is in theoretical mathematics, meaning it is done by using pen and paper instead of computing machines. Both Blair and Sogge took time to understand the pertinent literature on these problems and how best to approach this research.

The research contributes to the mathematical foundations of wave propagation. Being able to make a claim about standing waves informs much of what one can say about wave propagation altogether. “In some models, the high-frequency behavior of waves amplifies nonlinear interactions which lead to the formation of singularities and other chaotic phenomena. It turns out that examining Lp norms has shown to be useful towards understanding when this may or may not occur,” Blair said. 

“In the world of mathematics, our work deals with wave propagation on curved backgrounds described by non-Euclidean geometries or Riemannian manifolds. Here it is interesting to study how the underlying geometry and the paths of least action (geodesics) influence wave propagation. In the other direction, the standing waves on a Riemannian manifold reflect its geometry, so studying these helps us understand the geometric features of a Riemannian manifold.”

This research was supported in part by the National Science Foundation.