One of the reasons Allen Wu was interested in working at Brown was the strength the University has in both pure and applied mathematics. Wu’s research, in some ways, splits the difference between the two.
“I’m interested in applied math, but what excites me is using rigorous methods in pure mathematics to justify what we see in applied math,” said Wu, who joins the faculty as a Tamarkin Assistant Professor of Mathematics.
Much of his work so far has involved using partial differential equations to address problems in physics. One particular problem he’s been working on involves calculating the internal structure of rotating stars and gaseous planets.
“At the basic level, people are interested in figuring out the internal temperature, pressure, etc. of stars,” Wu said. “For instance, you can measure the radius of the Sun by a combined observation of distance and angular span, and you can measure the mass of the Sun by measuring the angular velocity of the Earth and using Newton’s gravity. By incorporating that with some modeling you can try to predict the internal temperature.”
But that’s not the hard part. Wu wanted to make those calculations more realistic by introducing two additional elements: rotation and, in the case of a planet, the possibility of a solid core.
“The addition of rotation and particularly the core kind of destroys the classical method for solving the equations,” Wu said. “So I had to come up with some different approach. Eventually, I was able to show that it’s doable.”
Another problem Wu has been working on involves what’s known as the Benjamin-Ono equation. The equation describes borders that form within fluids. For example, within oceans and lakes there are borders that form between water with low salinity and water with high salinity. In the atmosphere, such borders form when air masses of different density come together.
The equation describing these surfaces is extremely complex. One way to solve it is by essentially breaking it apart and reassembling it using what’s known as an inverse scattering transform. However, the use of that technique with this particular equation hadn’t been put on a firm footing.
“I was learning this framework and I didn’t quite understand it,” Wu said. “Later on, I found out that it wasn’t rigorously justified. So I started to work on that problem, and I made an initial step. I can justify a part of the scattering transform rigorously. So that’s a good start.”
Wu comes to Brown from Indiana University–Bloomington, where he was a visiting professor. He earned his Ph.D. in 2014 from the University of Michigan. He says he looks forward to exploring new questions and finding new problems to solve in collaboration with his new colleagues at Brown.
“I learn problems from people,” he said. “I’m not the kind of person who can feel which questions are important and just go for that. I talk to people and ask them what are the interesting questions in their area. I’m looking forward to starting new projects with the faculty here.”