July 3, 2025
PhD Alumni Ahmed Bou-Rabee was recently featured in Quanta Magazine for his joint work on Superdiffusion with Scott Armstrong (Courant) and Tuomo Kuusi (Helsinki). We asked Ahmed a few questions to learn more about the work and how his time at UChicago shaped his career. You can see the answers to those questions below.
1. Can you give us an overview of the research that was featured in Quanta and why it's significant in the field?
Watch a pollen grain in still water under a microscope and you’ll see a jittery, random dance — tiny thermal jolts cause it to move randomly, approximating a process called Brownian motion. A characteristic feature of Brownian motion is that its average squared displacement grows only linearly with time. Gently stir the water and the grain still dances randomly, but is also swept along by the eddies of the flow. This pairing of thermal noise and advection causes enhanced diffusion: the grain will explore space more quickly than it would through random motion alone.
Physicists predicted in the early 1980s that if the water is stirred randomly at just the right critical rate, this enhancement would cross into superdiffusion, that is, the mean squared displacement will grow superlinearly rather than linearly. This was compelling because it offered a simplified window into turbulence, one of the most notoriously complex phenomena in fluid dynamics.
Last year, Scott Armstrong, Tuomo Kuusi and I released a paper which proved this prediction. The significance of our work was not just in solving the problem, but also in the methods we introduced. We used ideas from quantitative stochastic homogenization, a field at the intersection of probability and partial differential equations, to build a rigorous framework for renormalization — a concept that physicists have used heuristically for decades.
2. How did you come to work on this problem with co-authors Scott Armstrong and Tuomo Kuusi?
I first came across Scott’s work during my PhD under Charles Smart — I read a lot of Scott’s papers and ended up applying some of his homogenization ideas in my thesis on the Abelian sandpile. This got me interested in homogenization, so I applied for a Courant Instructorship at NYU to work with Scott. I was fortunate enough to receive both the instructorship and the NSF postdoctoral fellowship, so I spent a year at Cornell and then the last 2 years of my postdoc at NYU. Tuomo is a long-time collaborator of Scott, and when I got to NYU, the superdiffusion problem was a natural direction for us to pursue together.
3. What were some of the biggest challenges you faced during your research?
Our proof is quite long, and there are a lot of dependencies in the arguments. Because of this, we had to make sure that all of the arguments were both air-tight and optimal. A suboptimal lemma in an early section can ripple forward fifty pages, costing weeks of time later. The challenge was in both mathematics and design: our proofs needed to be not just correct but robust, capable of supporting the weight of everything built upon them.
4. How has your experience as a PhD student at UChicago shaped your research and approach to science?
UChicago lives up to its reputation of intellectual intensity. It felt like the perfect place for me to learn and do research. Everyone was very friendly and it was humbling to be around leaders of the field everyday. The probability group especially was a pleasure to be around — I learned a ton from Steve Lalley and Greg Lawler. In my final year, I began a very fruitful collaboration with Ewain Gwynne, which continues to this day
5. What advice would you give students who are in the same position you were just a few years ago?
Don’t just pick an area of research because it’s very active, go into an area that you’re passionate about and willing to dedicate a lot of time to. Balance is also important; spend time thinking alone, but also talk to your fellow students and professors. I started a lot of successful collaborations from casual conversations.
6. What's next for you? Are there upcoming projects or questions you're excited to explore?
This fall I’m joining the University of Pennsylvania as a tenure track assistant professor. Scott, Tuomo and I have several follow up projects which we are actively working on which we hope will come to the arXiv soon! I’m also looking forward to advising PhD students and learning from them.
