Samuel Watson studies the mathematics behind two-dimensional statistical physics models. While he’s fascinated by the theorems and techniques used to study these models, what first got him interested were the images they produce.
“Researchers in the field draw some really amazing pictures using computer simulations,” said the new Tamarkin Assistant Professor of Mathematics. “The fractal curves we study, which look like lightning strikes, give you a visceral impression that the underlying models have rich mathematical structure.”
The models Watson works with aren’t necessarily geared toward simulating any specific physical systems. Rather, the dynamics are pared down to represent basic features of a more complex system. “The advantage of studying systems that are based on very simple rules and uncomplicated by layers of physical assumptions is that it’s possible to derive more information about the structure of the model in a rigorous way,” he said.
For a simple example, imagine hexagons in a honeycomb grid each colored blue or yellow according to independent coin flips. As it turns out, there are surprisingly predictable patterns that emerge from these systems that are driven essentially by randomness.
“If you trace out the boundary of a large cluster of blue cells,” Watson explained, “it turns out that you can say something quite precise about the length of this interface. Thus, while some features of the configuration will change from sample to sample, other features remain the same.”
The mathematical tools that have revealed this striking regularity in randomly generated systems are fairly new. The framework that unifies these models, known as the Schramm–Loewner evolution, was introduced in 1999.
“Because the research field is still young,” Watson said, “it enjoys a pretty rapid pace of new developments.”
In fact, research in the area has produced two recent Fields Medal winners.
At Brown, Watson hopes to learn a new approach to the subject from Brown mathematician Richard Kenyon.
“My work at MIT was primarily driven by analytic techniques,” Watson explained. “That means that quantities of interest are primarily studied through approximations, and exact results are obtained by taking limits. But Rick’s approach is different. He derives exact algebraic identities that describe the system precisely, and any desired approximations are derived from these identities. The research field at large has benefited quite a lot from the interplay between these points of view.”
The new job will also be a chance for Watson to become more acquainted with the top of College Hill. Last semester, he was a postdoctoral fellow at ICERM, Brown’s mathematics institute located in downtown Providence at the foot of College Hill.