Analyzing Approximations

Matthew Rosenzweig is a self-described late bloomer, mathematically speaking. Becoming a mathematician wasn't even on his radar as he began his undergraduate studies. That all changed when he took a course on linear algebra and real analysis.

"The course was really transformative," said Rosenzweig, an assistant professor of mathematical sciences. "It's what made me realize that I really liked proving theorems. I had trouble envisioning my life without that being a significant component."

Throughout his graduate work at the University of Texas at Austin and his postdoctoral research at the Massachusetts Institute of Technology, Rosenzweig gravitated toward problems that he could sink his teeth into. And when he found — or stumbled upon — those types of mathematical problems, he followed the research path they led him down.

"You get going in a direction, you start answering questions, they raise new questions, and you keep going down that path until you decide that you don't have any more interesting things to say. And I think that's a sign that you should branch off into a new direction," he said.

These days, Rosenzweig's research interests lie at the intersection of mathematical analysis, probability, and physics — an area quite different from his self-professed interests as a beginning graduate student. He focuses on non-linear partial differential equations (PDE) as asymptotic models of what should happen in physical systems, like waves on the ocean surface, vortices in a fluid, or charged particles in a gas.

These systems consist of a large number of interacting waves or particles — so large that describing the exact evolution of an individual wave or particle in such systems is intractable. Instead, mathematicians turn to an asymptotic approach, which gives an approximation of the system's typical behavior through the solution of a single PDE, which can be efficiently computed. The key question is: how good of an approximation is it? Rosenzweig works to understand the local and global dynamics of the asymptotic equation and how well the solutions approximate the underlying physics. While this is an applied question, answering it often leads to the development of new tools of pure mathematical interest.

Over the years, he has worked on topics including critical problems for dispersive PDE, effective equations and structures for classical and quantum many-body systems, and the effect of stochastic perturbations on PDE. As a postdoctoral researcher at MIT, he joined the Simons Collaboration on Wave Turbulence, which brings together mathematicians and physicists who study interacting wave systems. Working under a such an interdisciplinary umbrella inspired Rosenzweig to seek out people who approach things from completely different angles. It's one of the reasons he's happy to be at Carnegie Mellon.

"CMU is strong in areas where I would like to grow, and I'm looking forward to branching more outside of my mathematical comfort zone," he said.