Faculty Fanfare: Matt Blair

UNM Professor of Math Matt Blair received a Frontiers of Science Award at the inaugural International Congress on Basic Science in Beijing.

The award stems from his co-authorship on a paper that appeared in Inventiones Mathematicae  and is entitled “Logarithmic improvements in Lp bounds for eigenfunctions of the critical exponent in the presence of nonpositive curvature.” It was co-authored by Christopher Sogge of Johns Hopkins University, who accepted the award on the duo’s behalf in Beijing.

The paper explores so-called standing waves, which oscillate periodically in time about a fixed profile, Blair said. The waves are analogous to solutions of the stationary Schrödinger equation in quantum mechanics.  

“It turns out that standing waves provide a lens into much richer wave phenomena, i.e. the more we understand about standing waves, the more we can understand about wave propagation all together. Because of this, it is interesting to study the behavior of these standing waves at high frequency,” Blair said.

One place the standing waves are seen is in vibrating strings, where they are often called harmonics. 

“While vibrating strings are essentially a one-dimensional version of this phenomena, there are higher dimensional versions as well, such as vibrating drumheads and plates. In these cases, the amplitudes of these standing waves may grow as the frequency of vibration increases.  Moreover, the underlying geometry of the system affects how large these amplitudes can be, e.g. the standing waves generated by a circular drum may behave much differently than a square drum,” he said.

“Our work considered a means of quantifying the growth of such amplitudes as the frequency increases, known mathematically as an Lp norm.  In particular, we considered certain circumstances where the underlying geometry of the system yields light rays or paths of least action which behave chaotically. We showed that in these cases, the amplitudes of standing waves cannot grow as fast when compared to certain geometries that allow for light rays which are very stable and predictable.”

Blair said the paper has led to other work on the topic and he has worked with Sogge and others to consider what happens when certain irregularities are introduced into the system, in the form of a singular potential function.  He is currently working with his former Ph.D. student, Chamsol Park, to examine if the amplitude of standing waves can be large over a lower dimensional set, like a curve or a hypersurface.  

“Moreover, I am also looking at geometries which yield light rays exhibiting mixed behavior, partially chaotic, but in a sense partially stable as well. It is a mathematical area that continues to inspire me after years of study, and I don’t think I’ll be running out of problems to work on any time soon,” he said.