The Math of Universality
The growth of bacterial colonies, the piling up of snow during a storm, and the spread of a wildfire are random, seemingly unrelated events, yet they all follow universal mathematical laws. Caltech's new assistant professor of mathematics Lingfu Zhang wants to elucidate the math behind these growth patterns and understand how and why the math is so widespread.
Zhang studies probability theory, with a particular focus on an area called KPZ universality, where KPZ reflects the last names of the three scientists—Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang—who pioneered the theory and its defining equation in 1986.
"Even when these growth processes look quite different close up, when you zoom out, their overall shape and randomness often follow the same underlying laws," he says.
Zhang, who grew up in the Sichuan Province of China, earned two bachelor's degrees in mathematics and computer science from MIT in 2017. He earned a PhD in mathematics from Princeton University in 2022. Before joining Caltech in the summer of 2024, he served as a postdoctoral fellow at UC Berkeley.
We spoke with Zhang to learn more about his quest to find deeper relationships that link patterns in nature.
Were you interested in math when you were young?
I did Math Olympiad competitions in high school, but after I graduated, I was a little bit tired of math. I spent my first year of college doing architecture. After that, I spent another year learning computer science. By the end of college, I had rediscovered my passion for math. Today, I still enjoy this feeling of rigorous logic where you use your brain to solve problems in ways that are not possible with architecture or engineering. I like the gymnastics that take place in my brain.
What drew you to probability theory?
In probability, problems can be stated in a way that even first-year PhD students can understand. It's very clean and has lots of beautiful examples from real life. It's math but not extremely abstract.
Probability theory stems from the Renaissance and has its roots in work by the Italian polymath Girolamo Cardano who analyzed games of chance, such as gambling games involving cards or dice. Mid-20th century biologists and physicists got interested in it because it also turns out to be useful in many physical systems. For example, the KPZ universality, a family of models in probability theory, was first formulated by physicists to describe growth processes such as the growth of bacteria, smoke aggregates, flame fronts, tumors, etc.
But the theory remained quite mysterious to mathematicians for a long time. Beginning in 2000, mathematicians managed to accurately solve the probability distribution for some of these growth processes. Physicists had originally proposed a differential equation for the KPZ theory, but it turns out that the mathematicians used algebra, specifically representation theory, to solve the equation more precisely.
Can you explain what universality means in your field?
Universality is the notion that goes back to the very beginning of probability theory. It's about different systems exhibiting the same large-scale behaviors.
Back in the 16th or 17th century, the first theorem in probability had to do with something called Gaussian distribution, which you may know as the bell curve. Let's say you measure the heights of the members of a population and plot it out. The heights would be distributed randomly—they would have a Gaussian distribution. For a while, Gaussian was the only universal thing people knew about randomness. But later it was realized that many growth processes follow a different distribution, called the Tracy–Widom law, [which was discovered in 1994 by Craig Tracy and Harold Widom]. It's also random but not Gaussian. If you think of snow falling and piling up, for example, the snow interacts with itself, which makes the process more complex. This will follow a Tracy–Widom distribution.
It turns out that this Tracy–Widom law can apply to other contexts as well, such as social networks and the formation of traffic jams, and more. In these growth processes, the Tracy-Widom law is the probability distribution that solves the KPZ equation at one point, or location. So, if you were using the KPZ equation to describe falling snow, the random fluctuations in snow height at a given point would follow the Tracy–Widom law. In other words, Tracy–Widom law is a key facet of the more comprehensive KPZ theory; and historically it also played a pivotal role in its development in the past decades. I want to understand the math behind the universality of Tracy–Widom, and its appearance in KPZ.
What are some specific problems you have solved?
A work of mine mathematically proved the presence of Tracy–Widom laws in the running time of random sorting algorithms, and another set of my works developed techniques to prove Tracy–Widom laws in random matrices. Another intriguing problem I'm working on is to mathematically prove the observed Tracy–Widom law in more general growth processes. Take the growth of E. Coli colonies as an example—this has been solved assuming the bacteria reproduce themselves in certain particular ways, but the more general case remains a quite challenging problem, a version of the "strong KPZ universality conjecture."
How do you like Caltech so far?
I really like the small size. It's very convenient to talk to anyone. If you have requests or need resources, it's easy to find somebody even at a very high level. Also, the students here are brilliant. I might give them something to work on, and they will come back to me just days later, after making progress on tasks I thought would take months to complete!
I also like the emphasis on science and engineering here. My work connects to other disciplines such as physics and computer science, and the small size and close interpersonal connections mean there are more chances for interactions.
What do you love most about doing math?
Like I discovered in college, I feel that theoretical research is not limited in the same way that it can be in engineering or experiment studies. There are limitations to what can be achieved in real life, but in math, there are no hard barriers. It comes down to what you can imagine. I also like that math is universal. It's true on Earth and on any other worlds we might discover.
The accumulation of snow follows universal mathematical laws.
Credit: Shutterstock
